Geometry and Measurement ②: Extension of Trigonometric Ratios
We extend trigonometric ratios from 0° to 180°. They are defined using the unit circle.
Extended Definition
For a point P(cos θ, sin θ) on a circle of radius 1, they are also defined for 0° ≤ θ ≤ 180°:
sin(180° - θ) = sin θ, cos(180° - θ) = -cos θ, tan(180° - θ) = -tan θ
Example ①
Find sin 150°, cos 120°, and tan 135°.
Solution: sin 150° = sin 30° = 1/2. cos 120° = -cos 60° = -1/2. tan 135° = -tan 45° = -1.
Find sin 150°, cos 120°, and tan 135°.
Solution: sin 150° = sin 30° = 1/2. cos 120° = -cos 60° = -1/2. tan 135° = -tan 45° = -1.
Key Point
- sin is always positive from 0° to 180° (sin θ ≥ 0).
- cos is positive from 0° to 90° and negative from 90° to 180°.
- tan 90° is undefined.
Practice Problems
- Find sin 120°, cos 150°, and tan 120°.
Solutions and Explanations
- Solution: sin 120° = √3/2, cos 150° = -√3/2, tan 120° = -√3.