Law of Sines and Cosines

A. Law of Sines:

Given this triangle.

The Law of Sines is stated as

sin A = sin B = sin C ora b c

a = b = csin A sin B sin C

The area of the triangle equals one-half the product of the lengths of two sides and the sine of their included angle.

From the Law of Sines, it is observed that the sines of the angle sof a triangle are proportional to the lengths of the opposite sides.

Example:

In a Triangle ABC, a=4; b=4; A=75°Find B.

Solution: a = b sin A sin B

7/sin 75° = 4/sin B 4 in 75° = 7 sin B 4 (0.9659) = 7 sin B

sin B = 4(0.9659) / 7

Use logarithm in solving.

sin B = 0.5519 B = 34°

B. Law of Cosines:

On the other hand, the Law of Cosines may be stated as follows:

Given a triangle, the square of any length of any side equals the sum of the squares of the lengths of the other two sides decreased by twice the product of these two sides and the cosine of their included angle.

In symbols, a2 = b2 + c2 – 2bc cos Ab2 = a2 + c2 – 2ac cos B c2 = a2 + b2 - 2ab cos C

Example:

Given: In a triangle ABC, a = 3; b = 5; c = 60°

Solution:c2 = a2 + b2 - 2ab cos 60°c2 = 32 + 52 – 2(3) (5) (0.5)

= 9 + 25 – 15 = 34 – 15 = 19

c = 4.36