In △ABC, m∠A=15°, a=10, and b=11. Find c to the nearest tenth.

The law of cosine states that the square of any one side of a triangle is equal to the difference between the sum of squares of the other two sides and double the product of other sides and cosine angle included between them.

Let a, b, and c be the lengths of the three sides of a triangle and A, B, and C be the three angles of the triangle. Then, the law of cosine states that: a2 = b2 + c2 − 2bc·cosA. As stated above, the law of cosines in trigonometry generalizes the Pythagorean theorem.

Law of Cosines Formula

The law of cosines formula can be used to find the missing side of a triangle when its two sides and the included angle is given i.e., it is used in the case of a SAS triangle. We know that if A, B, and C are the vertices of a triangle, then their opposite sides are represented by the small letters a, b, and c respectively. The law of cosines formula is used to:

find a when b, c, and A are given (or)
find b when a, c, and B are given (or)
find c when a, b, and C are given (or)

find any angle of the triangle when a, b, and c are given.

There are three laws of cosines and we choose one of them to solve our problems depending on the available data.

a2 = b2 + c2 - 2bc·cosA
b2 = c2 + a2 - 2ca·cosB
c2 = a2 + b2 - 2ab·cosC

As,

m∠A=15°, a=10, and b=11.

Now, using cosine rule

cos A = b² + c² - a²/ 2bc

cos 15 = 121+ c²-100/22c

0.96 = 121+ c²-100/22c

21.25 c = 21 + c²

21 + c² - 21.25 c=0

By solving the quadratic equation

c=20.2 units.

Learn more about cosine rule here:

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