Using sin²θ+cos²θ=1 as the foundation, master the three fundamental relationships.
Three Formulas
- sin²θ+cos²θ=1
- tan θ = sin θ/cos θ
- 1+tan²θ = 1/cos²θ
Example ①
When sin θ = 3/5 (in the second quadrant), find cos θ and tan θ.
Solution:cos θ = −4/5 (in the second quadrant)、tan θ = −3/4
When sin θ = 3/5 (in the second quadrant), find cos θ and tan θ.
Solution:cos θ = −4/5 (in the second quadrant)、tan θ = −3/4
Example ②
When tan θ = 2, find cos²θ.
Solution:1+4 = 1/cos²θ → cos²θ = 1/5
When tan θ = 2, find cos²θ.
Solution:1+4 = 1/cos²θ → cos²θ = 1/5
Key Point
- Start with sin²+cos²=1, then determine the sign from the quadrant
Practice Problems
- When cos θ = −2/3 (in the third quadrant), find sin θ and tan θ.
- When sin θ+cos θ = √3/3, find sin θ cos θ.
- Calculate sin(π−π/3) + cos(π+π/4) − tan(−π/6).
Solutions and Explanations
- Answer:sin θ = −√5/3、tan θ = √5/2
- Answer:sin θ cos θ = −1/3
Explanation:(sinθ+cosθ)²=1+2sinθcosθ=1/3→sinθcosθ=−1/3。 - Answer:√3/2 − √2/2 + √3/3