Trigonometric functions are defined using the point P(cos θ, sin θ) on the unit circle. tan θ = sin θ / cos θ。
Signs in Each Quadrant
| Quadrant | sin | cos | tan |
|---|---|---|---|
| 1 | + | + | + |
| 2 | + | − | − |
| 3 | − | − | + |
| 4 | − | + | − |
Example ①
Find the following values: (1) sin(2π/3) (2) cos(5π/4) (3) tan(7π/6)
Solution:(1) √3/2 (2) −√2/2 (3) √3/3
Find the following values: (1) sin(2π/3) (2) cos(5π/4) (3) tan(7π/6)
Solution:(1) √3/2 (2) −√2/2 (3) √3/3
Example ②
Find θ when sin θ = −1/2（π ≤ θ ≤ 2π）。
Solution:θ = 7π/6, 11π/6
Find θ when sin θ = −1/2（π ≤ θ ≤ 2π）。
Solution:θ = 7π/6, 11π/6
Key Points
- sin²θ+cos²θ=1 / tan θ=sin θ/cos θ / sin is an odd function, and cos is an even function
Practice Problems
- Find sin(4π/3), cos(−π/3), and tan(3π/4).
- Solve cos θ = −√3/2（0 ≤ θ ≤ 2π）。
- Find sin(−π/6), cos(−π/6), and tan(−π/6).
Solutions and Explanations
- Answer:−√3/2、1/2、−1
- Answer:5π/6, 7π/6
- Answer:−1/2、√3/2、−√3/3