Trigonometry Laws of Sines and Cosines 3

Question 1

In triangle ABC, b = 20, c = 15, and angle A = 110°. Using the Law of Cosines, what is the value of a²?

Accepted Answer

a² = 400 + 225 − 600·cos 110°

Answer

a² = 400 + 225 + 600·cos 110°

Answer

a² = 400 − 225 − 600·cos 110°

Answer

a² = 625 + 600·cos 70°

Question 2

The Law of Sines produces the ambiguous case (SSA) when you are given angle A, side a, and side b. How many solutions exist if a = b·sin A?

Accepted Answer

Exactly one solution (a right triangle)

Answer

No solution

Answer

Exactly two solutions

Answer

Infinitely many solutions

Question 3

In triangle XYZ, x = 10, y = 14, z = 9. What is cos X?

Accepted Answer

(100 + 81 − 196) / 180

Answer

(196 + 81 − 100) / 252

Answer

(100 + 196 − 81) / 280

Answer

(81 + 196 − 100) / 252

Question 4

A triangular garden has sides of 40 ft, 55 ft, and 70 ft. Which expression gives the cosine of the largest angle?

Accepted Answer

(40² + 55² − 70²) / (2·40·55)

Answer

(55² + 70² − 40²) / (2·55·70)

Answer

(40² + 70² − 55²) / (2·40·70)

Answer

(40² + 55² + 70²) / (2·40·55)

Question 5

Angle A = 30°, side a = 4, side b = 8. How many valid triangles exist?

Accepted Answer

0 — no triangle possible

Answer

1 — exactly one triangle

Answer

2 — two triangles

Answer

Infinitely many triangles

Question 6

In a triangle, angle B = 45°, b = 6, a = 4. Using the Law of Sines to find angle A, what is sin A?

Accepted Answer

sin A = 4·sin 45° / 6

Answer

sin A = 6·sin 45° / 4

Answer

sin A = sin 45° / (4·6)

Answer

sin A = 4 / (6·sin 45°)

Question 7

A surveyor measures two sides of a triangular plot as 120 m and 95 m, with an included angle of 78°. What formula gives the third side?

Accepted Answer

c = √(120² + 95² − 2·120·95·cos 78°)

Answer

c = (120 + 95·cos 78°) / sin 78°

Answer

c = √(120² + 95²)

Answer

c = 120·sin 78° / sin 95°