AMC12 Trigonometry & Precalculus 3 — Questions and Answers
Question 1: Which expression equals cos^2(x) - sin^2(x)?
- sin(2x)
- 2cos(x)
- -cos(2x)
- cos(2x) (Correct answer)
Correct answer: cos(2x)
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?
- 10sin(60)/sin(75)
- 10/sin(75)
- 10sin(75)/sin(45)
- 10sin(45)/sin(75) (Correct answer)
Correct answer: 10sin(45)/sin(75)
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)?
- (sqrt(6)+sqrt(2))/4
- sqrt(3)/2
- sqrt(2)/2
- (sqrt(6)-sqrt(2))/4 (Correct answer)
Correct answer: (sqrt(6)-sqrt(2))/4
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)?
- 5
- 1
- 6
- -1 (Correct answer)
Correct answer: -1
tan(alpha+beta)=(2+3)/(1-2*3)=5/(-5)=-1.
Question 5: What is the inverse of f(x) = e^(2x)?
- ln(2x)
- 2ln(x)
- e^(x/2)
- (1/2)ln(x) (Correct answer)
Correct answer: (1/2)ln(x)
Setting y=e^(2x) and solving for x gives x=(1/2)ln(y), so f^(-1)(x)=(1/2)ln(x).
Question 6: How many solutions does sin(x) = sqrt(3)/2 have in [0, 2pi)?
- 1
- 4
- 0
- 2 (Correct answer)
Correct answer: 2
sin(x)=sqrt(3)/2 at x=pi/3 and x=2pi/3, giving 2 solutions.
AMC12 Trigonometry & Precalculus 3
Which expression equals cos^2(x) - sin^2(x)?