Find the angle of C

Question

Suppose we are given the three side lengths of a triangle: a=3, b=4, and c=square root of 13. Find the angle C opposite side c.

Answer 1

When you have 3 sides, or two sides and the included angle, you solve the triangle using the cosine law. Otherwise, when you have 2 sides and an angle or a side and 2 angles, you use the sin law.

You may write this as c^2 = a^2 + b^2 - 2ab cos C. However, in this case, we are trying to solve for c, so we rewrite

2ab cos C = a^2 + b^2 - c^2, or

cos C = (a^2 + b^2 - c^2)/(2ab), or

C = arccos (a^2 + b^2 - c^2)/(2ab) Substituting a = 3, b = 4, and c = sqrt(13), we get

C= arccos(3^2 + 4^2 - (sqrt(13))^2)/(2*3*4)

C = arccos(12/24) = arccos(1/2)

C = 60°

To see that this answer makes sense, notice that sqrt(13) is between 3 and 4, the other two sides.