This is a comprehensive set of exercises on radian measure, the unit circle, relationships among trigonometric functions, and graphs.
Example ①
When sin θ+cos θ = 1/2, find (1) sin θ cos θ and (2) sin³θ+cos³θ.
Solution:(1) sin θ cos θ = −3/8。(2) (1/2)(1+3/8) = 11/16
When sin θ+cos θ = 1/2, find (1) sin θ cos θ and (2) sin³θ+cos³θ.
Solution:(1) sin θ cos θ = −3/8。(2) (1/2)(1+3/8) = 11/16
Example ②
For y = A sin(Bθ) with period π and maximum value 3, find A and B.
Solution:A = 3、B = 2
For y = A sin(Bθ) with period π and maximum value 3, find A and B.
Solution:A = 3、B = 2
Review
- 180°=π rad / l=rθ / S=r²θ/2
- Relationships: sin²+cos²=1、1+tan²=1/cos²
- Graphs: amplitude |A|, period 2π/|B|
Practice Problems
- When sin θ−cos θ = √2/2, find sin θ cos θ.
- Solve sin θ = cos θ for 0≤θ≤2π。
- State the period, amplitude, and phase shift of y = 3sin(2θ+π/6).
Solutions and Explanations
- Answer:1/4
Explanation:(sin−cos)²=1−2sincos=1/2→sincos=1/4。 - Answer:θ=π/4, 5π/4
- Answer:Period π, amplitude 3, phase shift −π/12