In order to apply the law of cosines to find the length of the side of a triangle, it is enough to know which of the following ?the area of a triangle ?
The measure of an angle and the length of the side opposite that angle ?
the lengths of the two triangle sides and the measure of the angle between them ?
Solution:
In order to apply the law of cosines to find the length of the side of a triangle, we need to know the length of two sides of the triangle and the measure of the angle.
In trigonometry, the law of cosines relates the lengths of the sides of a triangle to the cosine of one of its angles.
As in figure below, the law of cosines states
c2 = a2 + b2 - 2ab cosγ
Also it can be written as:
a2 = b2 + c2 - 2bc cos⍺
b2 = c2 + a2 - 2ca cosꞵ
The law of cosines generalizes the Pythagorean theorem, which holds only for right triangles:
if the angle γ is a right angle (of measure 90 degrees, or π/2 radians), then cos γ = 0, and thus the law of cosines reduces to the Pythagorean theorem:
c2 = a2 + b2
For example: Given a triangle with the three sides: a = 8, b = 6 and c = 9. Find angle C.
cos C = (a2 + b2 − c2)/2ab
cos C = (82 + 62 − 92)/(2×8×6)
cos C = (64 + 36 − 81)/96
cos C = 19/96 = 0.19791
C = cos−1(0.19791) = 78.58°
In order to apply the law of cosines to find the length of the side of a triangle, it is enough to know which of the following ?
the area of a triangle ?
The measure of an angle and the length of the side opposite that angle ?
the lengths of the two triangle sides and the measure of the angle between them ?
Summary:
In order to apply the law of cosines to find the length of the side of a triangle, we need to know the length of two sides of the triangle and the measure of the angle.