Trigonometry Laws of Sines and Cosines 2

Question 1

In triangle ABC, a = 8, b = 11, and angle A = 35°. Using the Law of Sines, which equation correctly finds angle B?

Accepted Answer

sin B = 11 · sin 35° / 8

Answer

By the Law of Sines, sin B / b = sin A / a, so sin B = b · sin A / a = 11 · sin 35° / 8.

Question 2

A triangle has sides a = 5, b = 7, c = 9. Which formula correctly begins the calculation to find angle C using the Law of Cosines?

Accepted Answer

cos C = (a² + b² − c²) / (2ab)

Answer

The Law of Cosines gives cos C = (a² + b² − c²) / (2ab).

Question 3

In an obtuse triangle where angle B > 90°, the Law of Sines may produce an ambiguous case. When is angle B guaranteed to be obtuse?

Accepted Answer

When b is the longest side and b > a

Answer

If b is the longest side and b > a, then angle B must be the largest angle, guaranteeing it is obtuse.

Question 4

Using the Law of Cosines, what is the length of side c in a triangle where a = 6, b = 8, and C = 60°?

Accepted Answer

√52

Answer

c² = 36 + 64 − 2(6)(8)cos 60° = 100 − 96(0.5) = 100 − 48 = 52, so c = √52.

Question 5

Triangle PQR has angle P = 50°, angle Q = 80°, and side p = 12. What is the length of side q?

Accepted Answer

q = 12·sin 80° / sin 50°

Answer

By the Law of Sines, q / sin Q = p / sin P, so q = p · sin Q / sin P = 12 · sin 80° / sin 50°.

Trigonometry Practice Test