AMC12 Trigonometry 2 β Questions and Answers
Question 1: What does sin(A + B) equal?
- sinA cosB + cosA sinB (Correct answer)
- sinA cosB β cosA sinB
- cosA cosB β sinA sinB
- sinA sinB + cosA cosB
Correct answer: sinA cosB + cosA sinB
The sine angle addition formula is sin(A+B) = sinA cosB + cosA sinB.
Question 2: If sin(ΞΈ) = 3/5 and ΞΈ is in quadrant I, what is cos(ΞΈ)?
- 4/5 (Correct answer)
- 3/4
- 5/4
- 1/5
Correct answer: 4/5
By the Pythagorean identity, cos(ΞΈ) = β(1 β 9/25) = β(16/25) = 4/5.
Question 3: What is sin(2ΞΈ) if sin(ΞΈ) = 1/2 and ΞΈ is in the first quadrant?
- β3/2 (Correct answer)
- 1/2
- 1
- β2/2
Correct answer: β3/2
sin(2ΞΈ) = 2sin(ΞΈ)cos(ΞΈ) = 2 Γ (1/2) Γ (β3/2) = β3/2.
Question 4: In a triangle with sides a = 7, b = 8 and included angle C = 60Β°, side c equals:
- β57 (Correct answer)
- β113
- 9
- β85
Correct answer: β57
By the law of cosines: cΒ² = 49 + 64 β 2(7)(8)cos(60Β°) = 113 β 56 = 57, so c = β57.
Question 5: What is the exact value of cos(75Β°)?
- (β6 β β2)/4 (Correct answer)
- (β6 + β2)/4
- (β3 β 1)/2
- (β2 β β3)/2
Correct answer: (β6 β β2)/4
cos(75Β°) = cos(45Β°+30Β°) = (β2/2)(β3/2) β (β2/2)(1/2) = (β6ββ2)/4.
Question 6: For 2sin(x) = β2 on [0, 2Ο], how many solutions are there?
- 2 (Correct answer)
- 1
- 3
- 4
Correct answer: 2
sin(x) = β2/2 gives x = Ο/4 and x = 3Ο/4, exactly 2 solutions in [0, 2Ο].
AMC12 Trigonometry 2
What does sin(A + B) equal?