AMC12 Trigonometry & Precalculus 3

Question 1

Which expression equals cos^2(x) - sin^2(x)?

Accepted Answer

cos(2x)

Answer

The double angle identity states cos(2x)=cos^2(x)-sin^2(x).

Question 2

In triangle ABC with A=45 deg, B=60 deg, and c=10, what is side a by the Law of Sines?

Accepted Answer

10sin(45)/sin(75)

Answer

C=75 deg; a/sin(A)=c/sin(C) gives a=10sin(45)/sin(75).

Question 3

What is the exact value of cos(75 degrees)?

Accepted Answer

(sqrt(6)-sqrt(2))/4

Answer

cos(75)=cos(45+30)=cos45*cos30-sin45*sin30=(sqrt(6)-sqrt(2))/4.

Question 4

If tan(alpha)=2 and tan(beta)=3, what is tan(alpha+beta)?

Accepted Answer

-1

Answer

tan(alpha+beta)=(2+3)/(1-2*3)=5/(-5)=-1.

Question 5

What is the inverse of f(x) = e^(2x)?

Accepted Answer

(1/2)ln(x)

Answer

Setting y=e^(2x) and solving for x gives x=(1/2)ln(y), so f^(-1)(x)=(1/2)ln(x).

AMC12 - American Mathematics Competition 12 Practice Test