Law of Cosines Example Problem

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Law of Cosines Example Triangle

The Law of Cosines is a useful tool to find the length of a triangle’s side if you know the length of the other two sides and one of the angles. It is also useful for finding the internal angles of a triangle if the length of all three sides is known.

The Law of Cosines is expressed by the formula

a2 = b2 + c2 – 2bc·cos A

where the angle’s letter corresponds to the side across from the angle. The same is true for the other angles and their sides.

b2 = a2 + c2 – 2ac·cos B

c2 = a2 + b2 – 2ab·cos C

Law of Cosines – How Does it Work?

It is easy to show how this law works. First, let’s take the triangle from above and drop a vertical line to the side marked c. This divides the triangle into two right triangles with one common side of length h.

Law of Cosines triangle showing two right triangles formed by dividing the original triangle by its vertical.

For the yellow triangle,

x = b·cos A
h = b·sin A

The length of c was divided into two parts of length x and y.

c = x + y

solved for y:

y = c – x

Substitute the expression for x from above

y = c – b·cos A

Using the Pythagorean theorem for the red triangle:

a2 = h2 + y2

Substitute the equations for h and y from above to get:

a2 = (c – b·cos A)2 + (b·sin A)2

Expand to get

a2 = c2 – 2bc·cos A + b2·cos2A + b2·sin2A

Combine the terms containing b2

a2 = c2 – 2bc·cos A + b2(cos2A + sin2A)

Using the trig identity cos2A + sin2A = 1, this equation becomes

a2 = c2 – 2bc·cos A + b2(1)

a2 = c2 – 2bc·cos A + b2

Rearrange the terms to get the Law of Cosines

a2 = b2 + c2 – 2bc·cos A

The same technique can be used for the other sides to get the other two forms of this equation.

Law of Cosines Example – Find the Side

Find the length of the unknown side of this right triangle using the Law of Cosines.

I chose a right triangle for this example to make it easy to check our work. To find c using the Law of Cosines, use the formula

c2 = a2 + b2 – 2ab·cos C

On this triangle,
a = 12
b = 5 and
C = 90°

Plug in these values to get:

c2 = (12)2 + (5)2 – 2(12)(5)·cos 90°

c2 = 144 + 25 – 120·cos 90°

c2 = 169 – 120·(0)

c2 = 169 – 0

c2 = 169

c = 13

Let’s check this using the Pythagorean Theorem

a2 + b2 = c2

(12)2 + (5)2 = c2

144 + 25 = c2

169 = c2

13 = c

This agrees with the value we found using the Law of Cosines.

Law of Cosines Example – Find the Angles

Use the Law of Cosines to find the missing two angles A and B on the previous example’s triangle.

a = 12
b = 5
c = 13

Find A using

a2 = b2 + c2 – 2bc·cos A

(12)2 = (5)2 + (13)2 – 2(5)(13)·cos A

144 = 25 + 169 – 130·cos A

144 = 194 – 130·cos A

144 -194 = – 130·cos A

-50 = -130·cos A

0.3846 = cos A

67.38° = A

Since this is a right triangle, we can check our work using the definition of cosine:

cos θ = adjacent ⁄ hypotenuse

cos A = 5/13 = 0.3846

A = 67.38°

Find B using

b2 = a2 + c2 – 2ac·cos B

(5)2 = (12)2 + (13)2 – 2(12)(13)·cos B

25 = 144 + 169 – 312·cos B

25 = 313 – 312·cos B

25 – 313 = – 312·cos B

-288 = – 312·cos B

0.9231 = cos B

22.62° = B

Check again using the definition of cosine:

cos B = 12/13 = 0.9231

B = 22.62°

Another means of checking our work would be to make sure all the angles add up to 180°.

A + B + C = 67.38° + 22.62° + 90° = 180°

The Law of Cosines is a useful tool to find either a length or internal angle of any triangle as long as you know at least the length of two sides and one angle or the length of all three sides.

Science Notes Trigonometry Help

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