How to Get Other Angle Measurements When You Only Know One

Eugene is a qualified command/instrumentation engineer Bsc (Eng) and has worked every bit a programmer of electronics & software for SCADA systems.

Solving triangles

© Eugene Brennan

Trigonometry and the Basics of Triangles

In this tutorial, you'll learn virtually trigonometry which is a co-operative of mathematics that covers the relationship between the sides and angles of triangles.

We'll observe out about:

• Polygons and the Definition of a Triangle
• The Basic Facts About Triangles
• The Triangle Inequality Theorem
• Dissimilar Types of Triangles
• Using the Greek Alphabet for Equations
• Sine, Cosine and Tangent
• Pythagoras'southward Theorem
• The Sine and Cosine Rules
• How to Work Out the Sides and Angles of a Triangle
• Measuring Angles
• How to Calculate the Surface area of a Triangle

What Is a Triangle?

Past definition, a triangle is a polygon with three sides.

Polygons are aeroplane shapes with several directly sides. "Plane" just means they're apartment and ii-dimensional. Other examples of polygons include squares, pentagons, hexagons and octagons. The word plane originates from the Greek polús meaning "many" and gōnía meaning "corner" or "angle." So polygon means "many corners." A triangle is the simplest possible polygon, having only iii sides.

Polygons with different numbers of sides. Regular polgons have sides the same length.

© Eugene Brennan

Basic Facts About Triangles

• A triangle is a polygon with three sides.
• All the angles add up to a total of 180 degrees.
• The angle betwixt two sides can be anything from greater than 0 to less than 180 degrees.
• The angle between two sides can't be 0 or 180 degrees, because the triangle would and then get straight lines. (These are called degenerate triangles).
• Similar triangles have the same angles, but dissimilar length sides.

The Symbol for Degrees

Degrees can be written using the symbol º. So, 45º ways 45 degrees.

Angles of a triangle range from 0 to less than 180 degrees.

© Eugene Brennan

No matter what the shape or size of a triangle, the sum of the 3 angles is 180

© Eugene Brennan

Similar triangles take the aforementioned angles but different length sides.

© Eugene Brennan

What Is the Triangle Inequality Theorem?

This states that the sum of whatsoever two sides of a triangle must be greater than or equal to the remaining side.

What Are the Unlike Types of Triangles?

Before nosotros learn how to work out the sides and angles of a triangle, it's important to know the names of the different types of triangles. The classification of a triangle depends on two factors:

• The length of a triangle'south sides
• The angles of a triangle's corners

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Types of Triangles Past Sides and Angles

Y'all tin can allocate a triangle either past side length or internal angle.

Types of triangles by length of sides.

Blazon of Triangle by Lengths of Sides	Description
Isosceles	An isosceles triangle has two sides of equal length, and one side that is either longer or shorter than the equal sides. Angle has no bearing on this triangle blazon.
Equilateral	All sides and angles are equal in length and degree.
Scalene	All sides and angles are of different lengths and degrees.

Types of triangles past angle.

Blazon of Triangle by Internal Angle	Clarification
Right (correct angled)	1 angle is 90 degrees.
Astute	Each of the iii angles measure less than xc degrees.
Obtuse	One angle is greater than xc degrees.

Triangles classified by side and angles.

© Eugene Brennan

Using the Greek Alphabet for Equations

Another topic that we'll briefly encompass before we delve into the mathematics of solving triangles is the Greek alphabet.

In scientific discipline, mathematics, and engineering many of the 24 characters of the Greek alphabet are borrowed for use in diagrams and for describing certain quantities.

You may accept seen the character μ (mu) represent micro every bit in micrograms μg or micrometers μm. The capital letter Ω (omega) is the symbol for ohms in electrical engineering. And, of course, π (pi) is the ratio of the circumference to the diameter of a circle.

In trigonometry, the characters θ (theta) and φ (phi) are frequently used for representing angles.

Messages of the Greek alphabet.

© Eugene Brennan

How Do Y'all Find the Sides and Angles of a Triangle?

In that location are several methods for working out the sides and angles of a triangle. To observe the length or angle of a triangle, one can use formulas, mathematical rules, or the fact that the angles of all triangles add upwards to 180 degrees.

Tools to observe the sides and angles of a triangle

• Pythagoras's theorem
• Sine rule
• Cosine dominion
• The fact that all angles add upward to 180 degrees

Pythagoras'south Theorem (The Pythagorean Theorem)

Pythagoras's theorem uses trigonometry to discover the longest side (hypotenuse) of a right triangle (right angled triangle in British English language). It states that for a right triangle:

The square on the hypotenuse equals the sum of the squares on the other two sides.

If the sides of a triangle are a, b and c and c is the hypotenuse, Pythagoras's Theorem states that:

c ² = a ² + b ²

c = √(a² + b ²)

The hypotenuse is the longest side of a right triangle, and is located reverse the right bending.

And so, if you know the lengths of two sides, all yous take to exercise is square the ii lengths, add together the result, then take the foursquare root of the sum to go the length of the hypotenuse.

Pythagoras'southward Theorem

© Eugene Brennan

Example Problem Using the Pythagorean Theorem

The sides of a triangle are iii and 4 units long. What is the length of the hypotenuse?

Call the sides a, b, and c. Side c is the hypotenuse.

a = three
b = 4

c = Unknown

So, according to the Pythagorean theorem:

c² = a² + b ⁱⁱ

And so, c² = 3ⁱⁱ + 4² = 9 + 16 = 25

c = √25

c = 5

How Do You lot Measure out Angles?

You can apply a protractor or a digital angle finder like this one from Amazon. These are useful for DIY and construction if you need to measure an angle between 2 sides, or transfer the angle to another object. Yous can use this as a replacement for a bevel gauge for transferring angles e.g. when marking the ends of rafters before cutting. The rules are graduated in inches and centimetres and angles tin can be measured to 0.one degrees.

Note that this isn't suitable as a technical drawing instrument because the hub won't sit down apartment on paper unlike a protractor. Also since it'southward made of stainless steel, it has pointed corners which may be abrupt and therefore isn't suitable for young children.

You can draw and measure angles with a protractor.

© Eugene Brennan

Sine, Cosine and Tangent of an Angle

A right triangle has 1 angle measuring 90 degrees. The side opposite this bending is known as the hypotenuse (some other name for the longest side). The length of the hypotenuse can be discovered using Pythagoras's theorem, but to find the other two sides, sine and cosine must be used. These are trigonometric functions of an angle.

In the diagram below, one of the angles is represented by the Greek letter θ. (pronounced "the - ta"). Side a is known every bit the "reverse" side and side b is called the "adjacent" side because of their positions relative to the angle θ.

The vertical lines "||" effectually the words below hateful "length of."

And then sine, cosine and tangent are divers equally follows:

sine θ = |opposite side| / |hypotenuse|

cosine θ = |adjacent side| / |hypotenuse|

tan θ = |opposite side| / |adjacent side|

Sine, cosine and tan.

© Eugene Brennan

Sine and cosine apply to an bending, any angle, so information technology's possible to have ii lines meeting at a bespeak and to evaluate sine or cosine for that angle even though there's no triangle as such. Nonetheless, sine and cosine are derived from the sides of an imaginary right triangle superimposed on the lines.

For instance, in the second diagram above, the purple triangle is scalene not right angled. However, you tin imagine a right-angled triangle superimposed on the purple triangle, from which the opposite, adjacent and hypotenuse sides can be determined.

Over a range 0 to xc degrees, sine ranges from 0 to 1, and cosine ranges from one to 0.

Retrieve, sine and cosine only depend on the angle, non the size of the triangle. Then if the length a changes in the diagram in a higher place when the triangle changes in size, the hypotenuse c also changes in size, simply the ratio of a to c remains abiding. They are similar triangles.

Sine, cosine and tangent are often abbreviated to sin, cos and tan respectively.

The Sine Rule

The ratio of the length of a side of a triangle to the sine of the angle opposite is constant for all three sides and angles.

So, in the diagram below:

a / sine A = b / sine B = c / sine C

Now, you can check the sine of an angle using a scientific calculator or wait information technology up online. In the old days before scientific calculators, we had to look up the value of the sine or cos of an angle in a book of tables.

The reverse or reverse function of sine is arcsine or "inverse sine", sometimes written as sin^-1 . When you check the arcsine of a value, y'all're working out the angle which produced that value when the sine role was operated on it. And so:

sin (30º) = 0.v and sin ^-1 (0.5) = 30º

When should the sine dominion be used?

The length of one side and the magnitude of the angle contrary is known. Then, if any of the other remaining angles or sides are known, all the angles and sides can be worked out.

Example showing how to use the sine rule to calculate the unknown side c.

© Eugene Brennan

The Cosine Dominion

For a triangle with sides a, b, and c, if a and b are known and C is the included angle (the angle betwixt the sides), C can be worked out with the cosine dominion. The formula is as follows:

c = a ² + b ² - 2ab cos C

When should the cosine dominion be used?

1. You lot know the lengths of the two sides of a triangle and the included angle. Yous tin can so work out the length of the remaining side using the cosine rule.
2. You know all the lengths of the sides simply none of the angles.

And so, by rearranging the cosine rule equation:

C = arccos ((a ² + b ² - c ²) / iiab)

The other angles can be worked out similarly.

The cosine rule.

© Eugene Brennan

Instance using the cosine dominion.

© Eugene Brennan

How to Discover the Angles of a Triangle Knowing the Ratio of the Side Lengths

If you know the ratio of the side lengths, you tin use the cosine dominion to work out two angles then the remaining angle tin be found knowing all angles add to 180 degrees.

Example:

A triangle has sides in the ratio 5:7:8. Notice the angles.

Reply:

So call the sides a, b and c and the angles A, B and C and assume the sides are a = 5 units, b = seven units and c = viii units. It doesn't matter what the actual lengths of the sides are because all similar triangles have the same angles. So if we work out the values of the angles for a triangle which has a side a = 5 units, information technology gives us the issue for all these similar triangles.

Use the cosine rule. So c ⁱⁱ = a ² + b ² - 2ab cos C

Substitute for a,b and c giving:

8² = 5² + 7² - 2(5)(seven) cos C

Working this out gives:

64 = 25 + 49 - 70 cos C

Simplifying and rearranging:

cos C = i/7 and C = arccos(ane/7).

You can utilise the cosine rule again or sine dominion to find a second angle and the third angle can be found knowing all the angles add to 180 degrees.

How to Go the Surface area of a Triangle

In that location are three methods that can be used to detect the expanse of a triangle.

Method 1. Using the perpendicular superlative

The area of a triangle can be determined by multiplying half the length of its base by the perpendicular height. Perpendicular means at right angles. But which side is the base of operations? Well, y'all tin utilize any of the three sides. Using a pencil, you can work out the area by drawing a perpendicular line from one side to the opposite corner using a set square, T-square, or protractor (or a carpenter'south square if you're amalgam something). Then, measure the length of the line and use the following formula to get the area:

Area = 1/2ah

"a" represents the length of the base of the triangle and "h" represents the height of the perpendicular line.

Working out the area of a triangle from the base lengtth and perpendicular meridian.

© Eugene Brennan

Method ii. Using side lengths and angles

The simple method above requires you to actually mensurate the height of a triangle. If you know the length of 2 of the sides and the included angle, you can work out the area analytically using sine and cosine (see diagram beneath).

Working out the surface area of a triangle from the lengths of ii sides and the sine of the included angle.

© Eugene Brennan

Method 3. Utilise Heron'southward formula

All you demand to know are the lengths of the three sides.

Area = √(s(south - a)(due south - b)(s - c))

Where s is the semiperimeter of the triangle

due south = (a + b + c)/2

Using Heron's formua to work out the expanse of a triangle.

© Eugene Brennan

Summary

If you've made it this far, yous've learned numerous helpful methods to detect different aspects of a triangle. With all this information, y'all may be confused every bit to when you should employ which method. The table below should help y'all identify which rule to utilize depending on the parameters you accept been given.

Detect the Angles and Sides of a Triangle: Which Rule Do I Use?

A summary of how to work out angles and sides of a triangle.

Known Parameters	Triangle Type	Rule to Employ
Triangle is right and I know length of two sides.	SSS after Pythagoras's Theorem used	Use Pythagoras's Theorem to work out remaining side and sine rule to work out angles.
Triangle is right and I know the length of one side and one bending	AAS after third bending worked out	Use the trigonometric identities sine and cosine to piece of work out the other sides and sum of angles (180 degrees) to work out remaining angle.
I know the length of two sides and the angle between them.	SAS	Utilize the cosine rule to work out remaining side and sine rule to work out remaining angles.
I know the length of 2 sides and the angle opposite one of them.	SSA	Use the sine rule to work out remaining angles and side.
I know the length of one side and all three angles.	AAS	Use the sine rule to work out the remaining sides.
I know the lengths of all three sides	SSS	Utilise the cosine dominion in reverse to piece of work out each bending. C = Arccos ((a² + b² - c²) / 2ab)
I know the length of a side and the angle at each end	AAS	Sum of three angles is 180 degrees and then remainging angle tin exist calculated. Use the sine rule to work out the ii unknown sides
I know the length of a side and 1 angle		You demand to know more information, either one other side or one other bending. Thes exception is if the known angle is in a rightangled triangle and not the correct bending.

FAQs About Triangles

Below are some often asked questions virtually triangles.

What do the angles of a triangle add upwardly to?

The interior angles of all triangles add together up to 180 degrees.

What Is the hypotenuse of a triangle?

The hypotenuse of a triangle is its longest side.

What exercise the sides of a triangle add up to?

The sum of the sides of a triangle depends on the individual lengths of each side. Unlike the interior angles of a triangle, which always add together upward to 180 degrees

How do you calculate the area of a triangle?

To summate the area of a triangle, just utilise the formula:

Expanse = 1/twoah

"a" represents the length of the base of the triangle. "h" represents its tiptop, which is discovered past drawing a perpendicular line from the base of operations to the elevation of the triangle.

How practice you find the third side of a triangle that Is not right?

If you lot know ii sides and the bending between them, use the cosine rule and plug in the values for the sides b, c, and the angle A.

Side by side, solve for side a.

And then use the bending value and the sine dominion to solve for angle B.

Finally, use your cognition that the angles of all triangles add upwardly to 180 degrees to find angle C.

How exercise you detect the missing side of a right angled triangle?

Use the Pythagorean theorem to find the missing side of a triangle. The formula is as follows:

c² = aⁱⁱ + b ²

c = √(a ⁱⁱ + b ²)

What is the name of a triangle with two equal sides?

A triangle with two equal sides and one side that is longer or shorter than the others is called an isosceles triangle.

What is the cosine formula?

This formula gives the square on a side reverse an angle, knowing the angle between the other 2 known sides. For a triangle, with sides a, b and c and angles A, B and C the three formulas are:

a ² = b ² + c ² - iibc cos A

or

b ² = a ² + c ² - 2air conditioning cos B

or

c ^two = a ² + b ² - 2ab cos C

How to effigy out the sides of a triangle if I know all the angles?

You need to know at least one side, otherwise, yous tin can't work out the lengths of the triangle. In that location's no unique triangle that has all angles the aforementioned. Triangles with the same angles are similar simply the ratio of sides for whatever 2 triangles is the aforementioned.

How to piece of work out the sides of a triangle if I know all the sides?

Use the cosine rule in reverse.
The cosine rule states:

c ² = a ² + b ² - 2ab cos C

And so, past rearranging the cosine dominion equation, y'all can work out the angle

C = arccos ((a ⁱⁱ + b ² - c ²) / 2ab)

and

B

= arccos ((a ^two+ c ^two - b ²) / 2ac)

The third angle A is (180 - C - B)

How to notice the perimeter of a triangle

Finding the perimeter of a triangle is a straightforward operation. The perimeter is equivalent to the added lengths of all three sides.

perimeter = a + b + c

How to observe the height of a triangle

Finding the meridian of a triangle is easy if you have the triangle'due south expanse. If you're given the area of the triangle:

height = 2 x area / base

If you don't have the area, but only have the side lengths of the triangle, employ the following:

height = 0.5 ten √ ((a + b + c)(-a + b + c)(a - b + c)(a + b - c)) / b

If you only have two sides and the angle between them, effort this formula:

area = 0.5 (a)(b)(sin(γ)), and then

height = expanse(sin(γ))

Triangles in the Real World

A triangle is the almost bones polygon and can't be pushed out of shape easily, different a square. If you wait closely, triangles are used in the designs of many machines and structures because the shape is so strong.

The forcefulness of the triangle lies in the fact that when whatever of the corners are carrying weight, the side contrary acts equally a necktie, undergoing tension and preventing the framework from deforming. For case, on a roof truss, the horizontal ties provide forcefulness and prevent the roof from spreading out at the eaves.

The sides of a triangle can also act as struts, but in this instance, they undergo pinch. An example is a shelf bracket or the struts on the underside of an airplane wing or the tail wing itself.

How to Implement the Cosine Dominion in Excel

Yous can implement the cosine rule in Excel using the ACOS Excel role to evaluate arccos. This allows the included angle to exist worked out, knowing all three sides of a triangle.

Using the Excel ACOS role to work out an angle, knowing 3 sides of a triangle. ACOS returns a value in radians.

© Eugene Brennan

How to Calculate Arc Length of a Circle, Segment and Sector Area

This content is accurate and true to the best of the author'south knowledge and is not meant to substitute for formal and individualized communication from a qualified professional person.

Questions & Answers

Question: How do you find the remaining sides of a triangle if you have merely i angle and ane side given?

Answer: You lot need to have more than information. And so either ane side and the ii angles at each end or two sides and the angle between them.

Y'all can prove this to yourself by drawing out the single side and bending and seeing how you can describe as many different shaped triangles as y'all want.

Question: How do I notice the value if all three sides of a scalene triangle are unknown?

Answer: If all the sides are unknown, you can't solve the triangle. You need to know at to the lowest degree two angles and 1 side, or two sides and one angle, or one side and one bending if the triangle is a right-angled triangle.

Question: What is the formula for finding what an equilateral triangle of side a, b and c is?

Answer: Since the triangle is equilateral, all the angles are 60 degrees. Withal, the length of at least one side must exist known. Once y'all know that length, since the triangle is equilateral, you lot know the length of the other sides considering all sides are of equal length.

Question: How would you solve this problem: The angle of superlative of the top of a tree from point P westward of the tree is forty degrees. From a second point Q due east of the tree, the angle of elevation is 32 degrees. If the distance between P and Q is 200m, observe the height of the tree, correct to iv pregnant figures?

Respond: One angle is 40 degrees, the other angle is 32 degrees, therefore the tertiary angle opposite the base PQ is 180 - (32 + 40) = 108 degrees.

You know one side of the triangle has length PQ = 200 m

A right angled triangle is formed between point P, the pinnacle of the tree and its base and also bespeak Q, the peak of the tree and its base.

The best way to solve is to notice the hypotenuse of one of the triangles.

So use the triangle with vertex P.

Call the indicate at the meridian of the tree T

Phone call the height of the tree H

The bending formed between sides PT and QT was worked out as 108 degrees.

Using the Sine Dominion, PQ / Sin(108) = PT/ Sin(32)

And then for the correct angled triangle we chose, PT is the hypotenuse.

Rearranging the equation in a higher place

PT = PQSin(32) / Sin(108)

Sin(40) = H / PT

So H = PTSin(40)

Substituting the value for the hypotenuse PT nosotros calculated above gives

H = (PQSin(32) / Sin(108)) x Sin(twoscore)

= PQSin(32)Sin(40)/Sin(108)

= 71.63 m

Question: How practice I notice the missing side of a triangle when only its peak is known?

Respond: Utilise Pythagoras's Theorem. Add the sine, cosine and tan relationships betwixt angles and the hypotenuse of the triangle to work out the remaining side.

Question: How do you observe the length of all the sides of a right triangle if all y'all know is Cos B is 0.75?

Answer: Y'all can find the angle B from the arccos of 0.75 and then employ the fact that the iii angles add up to 180 to detect the remaining bending. All the same in that location is an infinite number of similar right triangles that have all three angles the same, so you demand to know at least the length of i side.

Question: Which formula is used when given ninety-degree triangle, contrary angle is 26 degrees and i leg is know?

Answer: Utilise the fact that the cos of an angle is the length of the side by side side divided past the hypotenuse, or the sine of an bending is the opposite side divided by the hypotenuse. In your case, you know the side opposite the angle.

And so sine (26 degrees) = length opposite side / length hypotenuse

Therefore

Length hypotenuse = length opposite side / sine (26 degrees)

Apply Pythagoras'south theorem to work out remaining side

and remaining bending = 180 - (90 + 26) = 64 degrees

Question: How do I find the angles of a triangle if I know the lengths of all three sides?

Answer: Use the cosine rule to observe one of the angles. You'll need to use the arccos or inverse cos function to work out the value of the angle. Then utilize the sine dominion to find some other angle. Finally, use the fact that the sum of the angles is 180 degrees to find the remaining third angle.

Question: What rule would be used to find the length of sides if all three angles are known?

Answer: At that place is an infinite number of similar triangles that take the aforementioned angles. Imagine if you take a triangle and you lot know all the angles. Yous tin go on making it bigger, but the angles stay the same. However, the sides go longer. And so you need to know the length of at least ane side. And so you tin can use the Sine Rule to work out the remaining three sides.

Question: How do you lot find the side of a right triangle given two angles and hypotenuse?

Reply: If you know two angles, so you can work out the third since all the angles sum to 180 degrees. If the sides are a, b and the hypotenuse is c (opposite angle A), and the angles are A, B and C, so Sin A = a/c, so a = cSin A. Also Cos A = b/c, then b = cCos A.

Question: ABC is a triangle in which AB=20 cm and bending ABC =30°.Given that the area of the triangle is xc cm^2, find the length of BC ?

Answer: The formula for the area of the triangle is (1/2)AB X BCSinABC

Then rearranging:

BC = area / (i/two)ABSin(ABC)

= 2area / ABSin(ABC)

Plug in the values to work out BC:

BC = ii x 90 / (20 x Sin 30)

Question: How do you solve the side lengths (given only their algebraic values - no numerical ones) and the 90 degree angle?

Answer: Employ the sine rule, cosine rule and Pythagoras theorem to express the sides in terms of each other and solve for the unknown variables.

Question: How practise y'all find an angle of an isosceles if yous only know two sides and the area?

Respond: Let the triangle have sides of length a, b and c and angles A, B and C.

Bending A is reverse side a

Angle B is opposite side b

Bending C is contrary side c

The ii equal sides are a and b and the bending between them is C

Area = (one/2)absinC

a, b and the area are known

So sin C = area / ((1/2)ab)

C = arcsin(expanse / ((one/2)ab))

A + B + C = 180

Only A = B

And so A + B + C = 2A + C = 180

And then A = (180 - C)/ii

Employ the cosine rule to find length c

Question: How practise I get the area of a scalene triangle if I take 2 sides and the angle between them?

Answer: Apply the formula i/2abSinC where a and b are the two sides and C is the angle between them.

Question: If I take a ane length of a triangle and the other angles how do I find the missing length using the sine method?

Answer: Telephone call the sides a, b and c and the angles A, B and C

a is known and likewise A, B and C

So the sine rule says that a/Sin A = b/Sin B and rearranging gives b = (a/Sin A)Sin B

Similarly a/Sin A = c/Sin C and rearranging gives c = (a/Sin A)Sin C

Question: What is the maximum and minimum value for the sine of an angle?

Answer: If θ is the angle, the maximum value of sine occurs when θ = 90 degrees or π/2 radians. The minimum value is -i and this occurs when θ = 270 degrees or 3π/2 radians.

Question: A greenhouse can be modeled as a rectangular prism with a one-half-cylinder on top. The rectangular prism is 20 feet wide, 12 feet high, and 45 feet long. The half-cylinder has a diameter of 20 feet. To the nearest cubic foot, what is the volume of the greenhouse?

Answer: The volume of the rectangular prism section is:

Length 10 width 10 height

= 45 x 20 x 12 = 10800 cubic anxiety

The volume of a cylinder is the cross-exclusive area x length

The cross-exclusive area is the area of a circle

Let R be the radius = 20/2 = x

and L be the length = 45

Expanse = πR²

Book = πR²L

For a one-half cylinder

Volume = πR²L/2

= 3.1416 (10)² ten 45/2 = 7069 cubic feet to the nearest cubic pes

Total book = 7069 + 10800 = 17869 cubic feet

Question: How do I know when to use the sine or cosine formula?

Reply: If you know the length of two sides and the bending between them, then yous tin utilize the cosine formula to work out the remaining side. Otherwise, the sine formula or Pythagorean theorem can be used.

Question: How should I arroyo the trouble - The triangles ABC and ACD are such that BC- 32 cm, Advertizing - 19cm, CD - 28cm BAC - 74 ( angle ) and ADC - 67 ( angle )?

Answer: Use the cosine rule to piece of work out Ac. Then the sine rule to work out the remaining angles/sides.

Question: How exercise I know when to use sine or cosine formula when given two degrees and ane length?

Respond: If the length is opposite one of the known angles, you can use the Sine Rule. If it isn't, y'all tin work out the third angle since the 3 angles sum to 180 degrees. Then use the Sine Rule. The Cosine Rule is normally used when you merely have 1 angle between ii known sides.

Question: Each of the equal angles in an isosceles triangle measures 36 degrees. What is the measure of the third angle?

Respond: All the angles in a triangle add up to 180 degrees. Both angles are 36 degrees and so that'southward 72 degrees. The remaining angle is 180 - 72 = 108 degrees.

© 2016 Eugene Brennan

Eugene Brennan (author) from Ireland on July 03, 2020:

How-do-you-do Jacob,

If you two angles, yous can summate the tertiary one considering all angles sum to 180 degrees. And then you demand at least one side length and y'all tin can use the sine rule to summate the others.

Jacob Halstead on July 03, 2020:

Finding lengths of a trisngle's sides using two base interior angles?

Eugene Brennan (author) from Republic of ireland on June 05, 2020:

Hi Swetha,

You need to know the length of at least one side. In that location are an infinite number of right angle triangles with the same three angles (like triangles).

If you know one side, you can use sine and cos to work out the other sides.

Swetha on June 05, 2020:

How to find iii sides when angles are given in a right angle triangle.Give a formula to solve it?

Eugene Brennan (author) from Ireland on June 02, 2020:

Hi Kayla,

Draw your triangle with the side 8cm as the base. Call this a.

Then draw side c at an bending of 45.5 to side a starting at the left of a. This is bending B. You don't know it's length, so just continue on the line

Draw side b starting at the right of the base a. You don't know the length of b either, so just continue it on to intersect side b.

Use method 2 above for expanse to first find the length of side c.

And so area = 1/ii air-conditioning sin B = one/two (eight) c sin 45.5 = 4c sin 45.5 = eighteen.54 square cm

Rearranging gives c = 18.54 / (4 sin 45.v)

When you work out this value for c, you can utilize the cosine dominion to find the length of the side b opposite the 45.5 degrees bending. Now you know the lengths of all the sides so you can use the sine rule to work out the angles.

Kayla on June 01, 2020:

Can yu please explain this question?

A triangle has one side length of 8cm and an adjacent angle of 45.5. if the area of the triangle is 18.54cm, summate the length of the other side that encloses the 45.5 bending

Cheers

Eugene Brennan (writer) from Ireland on May 13, 2020:

Hello,

And then call the sides a, b and c and the angles A, B and C and assume the sides are a = 5 units, b = 7 units and c = viii units. It doesn't matter what the actual lengths of the sides are because all similar triangles have the same angles. So if nosotros work out the values of the angles for a triangle which has a side a = 5 units, it gives u.s. the issue for all these similar triangles.

Use the cosine rule. So c² = a² + b² - 2abCos C

Substitute for a,b and c giving:

8² = 5² + 7² - 2(five)(seven)Cos C

Working this out gives:

64 = 25 + 49 - 70Cos C

Simplifying and rearranging:

Cos C = 1/7 and C = arccos(1/7).

Y'all tin utilise the cosine rule once more to discover a 2nd angle and the 3rd angle can be found knowing all the angles add to 180 degrees.

Hi on May 13, 2020:

Can I detect sinus of the biggest or the smallest angle, if the merely affair I know is that the triangle is astute and it'south sides are proportional to 5:7:8?

Eugene Brennan (author) from Republic of ireland on May 10, 2020:

Hi Abike,

No, because, there are an infinite number of combinations of angles for the other 2 angles or 2 sides.

Depict 2 lines with the known bending betwixt them. Yous'll come across that y'all tin can make the ratio of their lengths anything you want, changing the angles as well then that one is big and the other small-scale or vice versa.

Abike on May 10, 2020:

Hello,

Is information technology possible to find the angles of an acute triangle with just one known bending and no known side?

Eugene Brennan (author) from Ireland on April 29, 2020:

Apply the simple formula:

expanse = 1/2 the base x peak

Multiply both sides of equation by 2

2area = 2 10 one/ii 10 base x summit = base of operations by meridian

Split both sides by tiptop

2area/height = base 10 height/height = base of operations

and switch around the two sides

then base = 2area / height.

Suzy on April 28, 2020:

Find the length of the base. Where the height is viii and the area is xx. Solve for the length base of operations?

Emmy on Apr 07, 2020:

Thanks so much!

Himanshu gond india on March 12, 2020:

Thanks a lot sir

Eugene Brennan (author) from Ireland on Feb 27, 2020:

Howdy Hassan, if we don't know the length of the side c, we need to know an additional piece of information, the angle betwixt side a and b or 1 of the other angles.

Hassan on February 27, 2020:

Mr. Brennan, if nosotros have merely two side information for case a=five, b=ten, and nosotros know nothing about the angles and then how to calculate c and whatever angle. the triangle is non correct triangle.

Eugene Brennan (author) from Ireland on February 20, 2020:

No trouble Bob, glad to help! Have a great twenty-four hour period too!

Bob longnecker on February 20, 2020:

Mr. Brennan

Give thanks you very much. This is what I was looking for.

Have a bully day and best regards .

Bob L.

Eugene Brennan (writer) from Republic of ireland on February 20, 2020:

Hi Bob,

The length of the short side is 3.half-dozen" x tan(xxx) which works out at two.08" approx.

If the angle changes to 31 degrees, the short side is 3.6" x tan(31) = ii.16" approx.

So the length variation of the brusk side would vary with the tan of the bending. If yous expect at the graph of tan, in that location'due south an approximately linear variation up to about 45 degrees (and then the long side increases proportionately with the angle). Then the graph gets steeper at an increasing rate, and so the short side would change a lot for pocket-size variations of angle.

Bob longnecker on February 18, 2020:

The 3.6 side is opposite the lx° angle. The 3.6 side is the longest of the two short sides. I don't care about the hypotinuse. Merely desire to really see what a change in the thirty° bending does and how it affects the curt side. Kickoff I need the length of that side and and so the length of that side when I change the 30° angle to 31°. How much does ane° alter affect the length?

Eugene Brennan (author) from Ireland on February 18, 2020:

In your beginning problem Bob, which bending is the 3.half-dozen" length opposite? (or is this side the hypotenuse, the longest side?)

Bob longnecker on February 17, 2020:

Nonetheless trying only no luck!

Eugene Brennan (writer) from Republic of ireland on February 17, 2020:

You lot tin also use a triangle calculator similar this 1 and all you accept to do is input values for side length and angle. If yous have sufficient information, it will calculate the remaining sides and angles.

https://world wide web.figurer.internet/triangle-reckoner.htm...

Eugene Brennan (author) from Ireland on Feb 17, 2020:

If the triangle is correct angled, so:

sine (bending) = length of side contrary bending / length of hypotenuse

Therefore length of side opposite angle = length of hypotenuse 10 sine(angle)

Similarly cos (angle) = length of side side by side to angle / length of hypotenuse.

Therefore length of side adjacent to angle = length of hypotenuse 10 cos(angle)

Tan(angle) = length of side opposite angle/length of side adjacent.

So if you lot know all the angles (which you do), and one side, you can work out the remaining sides.

Bob longnecker on Feb 17, 2020:

Sorry to say I'grand 77 years old. I took trig and calc every bit a senior in high school "60" years ago. Learning information technology taught me how to think and problem solve in life dorsum then but never used information technology Perdue after that. Forgot what I learned dorsum and then.

Do have a valid reason for the answer just don't have the clothing with all to become back and learn trig once again.

What I really demand to know is how much B changes per degree of modify in the hypothesis. Example going from 30 to 31°how much increase in B length ? What is your calculated answer.

Sorry only as well tired to get back and visit 60 years ago when I was 17!

Thank y'all and all-time regards,

Bob longnecker

Eugene Brennan (author) from Ireland on February 17, 2020:

How-do-you-do Bob, you can utilise the sine, cos and tan relationships to work out problems like this.

Bob longnecker on February 17, 2020:

I have a triangle with angles of: 30,60 and 90°. Side A is know to exist 3.6". I want to know what brusque side B is. Can anyone give me the answer?

Problem #2.

I have a triangle with angles of 31, 59, and 90°. Long side A is 3.half dozen". I desire to know the length of short side B.

Hi on Feb 12, 2020:

solve 2 triangle and 4 triangle in quadrilateral by utilise of sine rule

a/sin A= b/sinB that i know

Only

a = sin A/ sinB what is that formula

I don't understand that formula only that true

Duran on January 22, 2020:

Hello Mr. Brennan.

I have a trouble that is hard for me:

Known:I take 2 angles:∠A and ∠B and then I have a bundle of similar triangle-ABCs. At present there must be a indicate T inside the triangles who forms three new sides: TA, TB and TC. I know that the angles between all these three sides are every bit 120 deg.

Q: Can I solve the bending-BAT.

That realy confused me for a while !

Eugene Brennan (author) from Ireland on January 04, 2020:

If the angle is 45 degrees, the remaining angle is also 45 degrees, so the triangle is isosceles as well as being right angled. So if the length of the hypotenuse is a and the other 2 sides are b and c, then from Pythagoras's theorem:

a^ii = (b^2 + c^ii) = (2b^2)

and so b^2 = (a^ii)/2

and b = c = a / square root of 2

Nathaniel Gloyd on January 04, 2020:

If you have a correct angle triangle, how would you detect the distance from the corner of the 90 degree, to the hypotenuse on a 45 caste angle

Eugene Brennan (author) from Ireland on December 19, 2019:

How-do-you-do Rj,

Use the sine rule.

So if your sides are a,b and c and you know their lengths and your angles are A, B and C and yous know i angle A, then:

a/sin A = b/sin B

Plow both sides of the equation upside down, so:

sin A / a = sin B / b

Multiply both sides past b

b sin A / a = sin B

Work out b sin A /a on your calculator and this gives yous sin B.

Then accept the arcsin of the issue to get B. In one case you have A and B, add together and subtract from 180 to go C.

Rj on December xix, 2019:

If one bending and all 3 sides of the scalane triangle is given then how volition you get the measure of

other two angle

Eugene Brennan (author) from Ireland on Oct 24, 2019:

Hi Natalia,

Look at method 2 in the tutorial for finding the expanse of a triangle.

And so the surface area is 1/2 the product of two sides multiplied past the sine of the angle between them.

In your question the sides are PQ and QR and the bending between them is PQR.

And so area = (one/2) PQ sin PQR

Substitute for P, Q, angle PQR and the area:

14.2 = (ane/ii) x 7 ten 5 10 sin PQR

Rearrange:

sin PQR = 14.ii / ( (1/2) 10 7 x 5 )

Take the arcsin of both sides. You can do all this on a calculator, only have care entering all the brackets and numbers because it'southward very piece of cake to make a mistake. Make sure the calculators is set to "DEG" and use the sin ^ -1 (usually shift on sin) to work out arcsin.

I would recommend HiPer Calc equally a skilful, gratis scientific calculator app for Android if you have a smartphone.

PQR = arcsin (fourteen.two / ( (ane/2) x seven 10 5 ) ) = 54.235° = 54° xv' approx

natalia on Oct 24, 2019:

HI EUGENE, tin you lot solve this problem for me and provide me with working out.

the area of triange PQR is 14.2cm squared, find bending PQR to the nearest minute, given PQ is 7cm and QR is 5cm.

Eugene Brennan (author) from Ireland on Oct 09, 2019:

Hi Pavel,

By diagonal, I assume yous hateful the hypotenuse.

So you can apply Pythagoras' Theorem.

The foursquare on the hypotenuse equals the sum of the squares on the other ii sides.

Square the two sides and add together together:

(n + 4)² + xvi² = (north + 8)²

Aggrandize out:

n² + 8n + 16 + 256 = northward² + 16n + 64

Rearrange and simplify:

8n = 208

Giving northward = 26

So the two sides are n + 4 = 30 cm and n + 8 = 34 cm

Pavel on October 09, 2019:

I have a trouble well-nigh a question can you help me delight?

I have a right angled triangle the bottom line is xvi cm the one on the side is n+4 and the diagonal line is n+8 can you lot help me observe the ii sides delight?

Eugene Brennan (author) from Ireland on September 28, 2019:

Hi Carcada. Yous tin't. Y'all can have as many triangles as you desire with exactly the same three angles. These are called similar triangles. You need to know at least the length of 1 side, and so you lot can utilise the sine rule to work out the others.

Carcada Keischa on September 28, 2019:

if just the angles of each side of the triangle is given and then how can we find the length of each side of the triangle?

Eugene Brennan (writer) from Ireland on September 08, 2019:

You don't have enough information. Y'all demand to have at least one of a, c, A or C.

Sin B = ane/ sqrt 3, merely gives you the angle B = (acos (i/sqrt iii)). So if a is the base, side c can be any length without knowing the other sides/angles.

Hannah Adams on September 07, 2019:

I have a question. How practise I find the missing sides of a triangle if I know that sin B=1/sqrt 3 and a=ii

Eugene Brennan (author) from Republic of ireland on August 14, 2019:

tan (ɵ) = reverse / side by side so opposite = adjacent x tan (ɵ)

Now you lot know the opposite and adjacent sidfes, use Pythagoras' theorem to work out the hypotenuse.

Phoebe on August 13, 2019:

Hey, i take a triangle, all that is known is the adjacent, the correct angle and the theta, how exercise i figure out the other sides,

asaba charles on July 23, 2019:

thanks

Maribel Gibbs from Paoli, Pennsylvania on May 22, 2019:

Wow, amazing! One of the best works I ever take seen here!

Khaleel Yusuf on May xviii, 2019:

A good review of many years of wining and dining with math calculations. Awesome!

Ur mum gay on Apr 24, 2019:

This is a decent website

Christopher on March 26, 2019:

Wow this is actually helpful thanks

Michael on January 20, 2019:

Hi,

I'm wrapping my head around this trouble: I know i side, and the two angles produced past the median on the opposing corner. I'd like to know the length of the other two sides. I drew a scheme, available here:

www.Stavrox.com/image/Triangle.png

The green values are known (a, alpha, beta) , I'd like to calculate b, c and as well x. Can you lot assistance me.

Ferny Vise from San Francisco, CA on January nineteen, 2019:

I really like this commodity. As a math major myself, I believe math is cute!

Oscar Skabar on December 02, 2018:

I have an example I cannot piece of work out..... Two birds sitting on a ninety degree mask one at 9m up & the other at 6m upward simply are 15m apart from each other, they run into a fish in the water, how practise I calculate the distance of the fish from the birds so they are equal in distance

Rodrigo on November xix, 2018:

Hi, Eugene! Yous can calc the three angles inside a triangle using tangent one-half-bending like this:

tan(alpha/2) = r / (p-a)

tan(beta/two) = r / (p-b)

tan(gamma/2) = r / (p-c)

p = (a+b+c) / 2 (semiperimeter)

r = sqrt( (p-a)(p-b)(p-c) / p )

alpha + beta + gamma = 180 (they are the internal angles of the triangle :)

Congrats for your site!

Eugene Brennan (writer) from Ireland on Nov eighteen, 2018:

Hello Carla. There may be a simpler way of doing it, but you can use the cosine rule in opposite to work out the angle B. So since it'south bisected, y'all know half this angle. And so use the cosine rule in reverse or the sine dominion to work out the bending between sides AB and CA. You know the third angle (between the bisector line and side CA) considering the sum of angles is 180 degrees. Finally utilise the sine rule once more to work out the distance from A to the bisection point knowing the length of AB and half the bisected angle.

Eugene Brennan (author) from Republic of ireland on November 05, 2018:

Yous can't find side lengths with angles alone. Similar triangles have the same angles, merely the sides are unlike. You must have the length of at least i side and 2 angles.

william on November 05, 2018:

how exercise you lot find side lengths with only angle measurements

Eugene Brennan (writer) from Republic of ireland on November 03, 2018:

If yous have the bending at each end, so you tin work out the third bending because you know all the angles add up to 180 degrees. Then employ the sine rule to piece of work out each side (meet example above in the text)

Karen on November 02, 2018:

i have the the length of ane side and the angle at each end, what is the sum to work out the length of the other sides

Eugene Brennan (author) from Ireland on Nov 02, 2018:

Hi Tom,

If you lot know the lengths of all three sides, utilize the cosine dominion get-go and the arccos function to work out one of the angles. Then utilize the sine rule (or the cosine rule again) to work out the one of the other two angles and the fact that they add together up to 180 degrees to observe the terminal bending

As regards Excel, I've added a photograph to the article showing how to implement a formula for working out an angle using the cosine rule.

tom sparks on Oct 23, 2018:

I take a correct angled triangle and know the lengths of all iii sides. I would like to calculate the other angles.

I have tried TAN in Excel but information technology says using this 'Returns the tangent of the given bending,.

What would exist the best way to work this out

Hope y'all can help

Kind regards

Eugene Brennan (author) from Ireland on October 21, 2018:

You need more than information, either another side or angle to solve.

Sanjeev on October 21, 2018:

Right bending and h is 421.410

How find 2 angles and 2 sides.

Eugene Brennan (author) from Ireland on September 28, 2018:

You lot kneed to know at least one other angle or length. The exception is a right-angled triangle. If you know one bending other than the right angle, so you can piece of work out the remaining angles using sine and cos relationships betwixt sides and angles and Pythagoras' Theorem.

SUDHAKAR G on September 28, 2018:

how to i find the length in a Scalene triangle? we konw merely one angle and i length.

Eugene Brennan (author) from Ireland on August 25, 2018:

If ii sides are given and the angle between them, use the cosine rule to notice the remaining side, then the sine rule to find the other side.

If the angle isn't between the known side, employ the sine rule to discover the angles outset, and then the unknown side.

Y'all at least need to know the angle betwixt the sides or one of the other angles then in your example information technology'south the sine rule you lot need to utilise.

Akhyar on August 24, 2018:

If only 2 sides are given of a not right angled triangle .. and then how to notice bending betwixt them

Eugene Brennan (author) from Ireland on July 19, 2018:

Hi Imran,

There's an infinite number of solutions for angles A and B and sides a and B. Depict it out on a piece of newspaper and yous'll see that you can orientate side c with a known length (due east.yard. choice a length of ten cm) and change the angles A and B to what always you want.

You need to know either the length of one more side or one more bending.

Imran Hussain from Bharat on July nineteen, 2018:

Call the angles A,B and C and the lengths of the sides a, b and c.

a is contrary A

b is opposite B

c is opposite C

C is the right angle = 90º and c is the hypotenuse.

How to detect the sides of triangle a and b and other ii angles A and B, if i know just angle C and side c which is hypotenuse?

Eugene Brennan (author) from Ireland on May 28, 2018:

Hi Liam,

Y'all demand to know at to the lowest degree ane of the sides.

You could have a very large or very small triangle with the same angles. These are chosen similar triangles. Encounter the diagram in the tutorial.

Liam on May 27, 2018:

How do I find a side in a right bending triangle if I know all 3 angles but no sides?

Eugene Brennan (author) from Ireland on May 24, 2018:

If the holes are every bit spaced around the imaginary circle, then the formula for the radius of the circle is:

R = B / (2Sin(360/2N))

Where R is the radius

B is the distance between holes

N is the number of holes

Divya on May 24, 2018:

how to summate distance of each pigsty at PCD from eye circle

Amar36 on April 17, 2018:

Hi sir

how is that possible to know angle by just having ratios of 2 heights of triangle and u need non utilise protector or some other instruments and not even inverse trigonometric functions but simply by ratio do we summate them or not if then how

I asked information technology because how they accept founded the angles of different triangles with it any discovery of inverse trigonometric functions.

Give thanks in advance

Eugene Brennan (author) from Ireland on February 13, 2018:

No enough information shahid! If you think about it, at that place'south an infinite number of triangles that satisfy those conditions. Area = (1/2) base x height. So in that location's no unique values of base and height to satisfy equation (ane/2) base x height = 10 m squared.

shahid abbasi on February 13, 2018:

area of right angle triangle is 10m and one bending is 90degree then how summate three sides and another two angles.

Eugene Brennan (author) from Ireland on January 14, 2018:

If yous assign lengths to all sides, y'all easily tin can piece of work out the angles. Which sides did assign a length to?

Gem on Jan 13, 2018:

Any luck Eugene? I have figured out some of the angles by folding a function of the paper that can permit me use trig to figure it out if I assign each side a length.

Eugene Brennan (author) from Ireland on January 07, 2018:

Hi Danya,

Because you know two of the angles, the tertiary angle can simply be worked out past subtracting the sum of the ii known angles from 180 degrees. And so use the Sine Dominion described above to work out the two unknown sides.

danya61 on Jan 07, 2018:

Hello

I have a triangle with ii known angles and one known length of the side betwixt them, and there is no right bending in the triangle. I want to calculate each of unknown sides. How can I do that? (The angle between unknown sides is unknown.)

Eugene Brennan (author) from Ireland on January 04, 2018:

Draw a diagram jeevan. I can't really visualize this.

jeevan on Jan 04, 2018:

there are 3 circles 1 big circle is a pitch circle having 67 diameter and medium circle is fatigued on the circumference of pitch circumvolve at the angle of five degree hvaing 11.04 radius and a small circle with only moves in x y direction on pitch circle radius having i.5 radius so if the medium circle is moved 5degree then at which point the small circle is coinciding and the distance from small circle to center of large/pitch circle.?

sir please help me finding the answer give thanks yous.

Gem on December 29, 2017:

It is tough to show for sure. I thought I had it past assigning each side a random length ( such as 2cm) and and then taking the center point every bit one-half, which looked like the correct angle triangle on the superlative right hand side was half of the one-half. But it still tin can't be proven to exist half considering of the fold.

Eugene Brennan (author) from Ireland on Dec 16, 2017:

If it's an equilateral triangle, the sides and angles can be easily worked out. Otherwise the triangle can have an infinite number of possible side lengths as the apexes A and C are moved around. So if none of the magnitudes of lengths are known, the expression for lengths of sides of the triangle and its angles would accept to be expressed in terms of the square's sides and the lengths AR and CP?

Precious stone on December 15, 2017:

The whole problem has no measurements or angles. Information technology only has bending names such as A,B,C,D etc. My starting point is from the common noesis that a square has four x ninety degree angles. If I could determine i other angle then I could effigy out the whole problem by using the 180 degree rule of triangles. I will snap a picture of it and endeavor and upload information technology here on Monday, or sketch and upload it. It seems to be a real stumper, two/lxx people at a workshop were able to figure it out, as I was told by the person who passed it along to me. I appreciate your answer, and I look forward to sharing the appropriate visual information with y'all.

Eugene Brennan (writer) from Ireland on December fifteen, 2017:

Howdy Gem,

Is any information given about where the corners of the triangle touch the sides of the square or the lengths of the square's sides? If the triangle isn't equilateral (or even if it is), it seems that there would exist an infinite number of placing the triangle in the foursquare.

Precious stone on Dec xiv, 2017:

Problem: A triangle is placed inside a square. The triangle doesn't have measurements or any listed angles. So we can't place the blazon (although it looks equilateral) or brand any physical assumptions virtually the triangle. I'm suppose to figure out the angles of the triangle without a protractor or ruler based on the only angles I am given which are the 90 degrees from each corner of the square it's in. Since the lines that cut through the foursquare from the main triangle within the square make new sets of smaller triangles, I notwithstanding tin can't make out complimentary or supplementary angles since virtually of those smaller triangles aren't definitely correct angles isosceles triangles.

I'grand not certain if my question is clear, and so if you reply back I'll try and add a film or sketch to analyze.

Merely motion picture a square with a triangle in it touching all iii sides of its points to the square with no units of measure and no angles. We tin only assume that the square has 90 degree angles in the corners and that's all we are given to work with.

Thanks Precious stone

Eugene Brennan (author) from Ireland on December 01, 2017:

Hullo Maxy,

Call the angles A,B and C and the lengths of the sides a, b and c.

a is reverse A

b is opposite B

c is contrary C

C is the correct angle = 90º and c is the hypotenuse.

If the bending A is known and the side opposite information technology, a, is known

And so Sin A = opposite/hypotenuse = a/c

So c = a/Sin A

Since you lot know a and A, you can work out c.

So use Pythagoras's theorem to work out b

c² = a² + b²

So b² = c² - a²

And so b = √(c² - a²)

If the angle A is known and the side adjacent to information technology, b, is known

Then Cos A = adjacent/hypotenuse = b/c

So c = b / Cos A

Since you know b and A, you tin can work out c.

Then apply Pythagoras'southward theorem to work out a.

c² = a² + b²

So a² = c² - b²

And so a = √(c² - b²)

Eugene Brennan (author) from Ireland on November 27, 2017:

You lot need to use the cosine rule in reverse.

And so if the angles are A, B, and C and the sides are a,b and c.

Then c² = a² + b² - 2abCos C

Rearranging gives angle C = Arccos ((a² + b² - c²) / 2ab)

You tin can work out the other angles similarly using the cosine rule. Alternatively use the sine dominion:

So a/Sin A = c/Sin C

Then Sin A = a/c (Sin C)

and A = Arccos ( a/c (Sin C) )

and similarly for the other angles

Hannah on Nov 27, 2017:

How do you notice the angle if all three sides are given

Eugene Brennan (author) from Ireland on November 25, 2017:

Polygons are a lot more complicated than triangles considering they can take whatever number of sides (they do of form include triangles and squares). Also polygons can exist regular (have sides the same length) or not-regular (have different length sides).

Here's two formulae:

For a regular or non-regular polygon with northward sides

Sum of angles = (n-ii) x 180 degrees

For a regular convex polygon (not like a star)

Interior angles = (ane - 2/n) x 180 degrees

Eugene Brennan (writer) from Ireland on November 23, 2017:

How-do-you-do Jeetendra,

This is called a scalene triangle. The longest border of whatever triangle is reverse the largest angle. If all angles are known, the length of at least i of the sides must exist known in order to detect the length of the longest edge. Since you know the length of an edge, and the angle contrary it, you can utilize the sine dominion to piece of work out the longest edge. So if for instance you know length a and angle A, and then yous can piece of work out a/Sin A.

If c is the longest side,

then a/sin A = c/Sin C ,

so rearranging,

c = a Sin C / Sin A

a, C and A are known, so yous tin work out c

Jeetendra Beniwal( from Bharat) on November 23, 2017:

If all three angles are given then how we find largest border of triangle,if all angles are astute

Eugene Brennan (author) from Republic of ireland on July 21, 2016:

Cheers Ron, triangles are smashing, they crop up everywhere in structures, machines, and the ligaments of the human torso tin be idea of equally ties, forming 1 side of a triangle.

Ron Bergeron from Massachusetts, US on July 21, 2016:

I've always establish the math behind triangles to exist interesting. I'yard glad that yous ended the hub with some examples of triangles in every day use. Showing a practical use for the information presented makes it more interesting and demonstrates a purpose for learning about it.

smithspoing.blogspot.com

Source: https://owlcation.com/stem/Everything-About-Triangles-and-More-Isosceles-Equilateral-Scalene-Pythagoras-Sine-and-Cosine

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