AMC12 Trigonometry 2

Question 1

What does sin(A + B) equal?

Accepted Answer

sinA cosB + cosA sinB

Answer

The sine angle addition formula is sin(A+B) = sinA cosB + cosA sinB.

Question 2

If sin(θ) = 3/5 and θ is in quadrant I, what is cos(θ)?

Accepted Answer

4/5

Answer

By the Pythagorean identity, cos(θ) = √(1 − 9/25) = √(16/25) = 4/5.

Question 3

What is sin(2θ) if sin(θ) = 1/2 and θ is in the first quadrant?

Accepted Answer

√3/2

Answer

sin(2θ) = 2sin(θ)cos(θ) = 2 × (1/2) × (√3/2) = √3/2.

Question 4

In a triangle with sides a = 7, b = 8 and included angle C = 60°, side c equals:

Accepted Answer

√57

Answer

By the law of cosines: c² = 49 + 64 − 2(7)(8)cos(60°) = 113 − 56 = 57, so c = √57.

Question 5

What is the exact value of cos(75°)?

Accepted Answer

(√6 − √2)/4

Answer

cos(75°) = cos(45°+30°) = (√2/2)(√3/2) − (√2/2)(1/2) = (√6−√2)/4.

AMC12 - American Mathematics Competition 12 Practice Test