The ratios of the sides of a right triangle are called trigonometric ratios (sin, cos, and tan). Their values are determined by the size of the angle.
Definitions of Trigonometric Ratios
In a right triangle, let the hypotenuse be c, the side opposite angle θ be a, and the adjacent side be b:
sin θ = a/c, cos θ = b/c, tan θ = a/b
Example ①
Verify that sin 30° = 1/2, cos 30° = √3/2, and tan 30° = 1/√3 = √3/3.
Solution: This can be confirmed from a 30-60-90 triangle (side ratio 1:√3:2).
Verify that sin 30° = 1/2, cos 30° = √3/2, and tan 30° = 1/√3 = √3/3.
Solution: This can be confirmed from a 30-60-90 triangle (side ratio 1:√3:2).
Commonly Used Trigonometric Ratios
| θ | 30° | 45° | 60° |
| sin | 1/2 | √2/2 | √3/2 |
| cos | √3/2 | √2/2 | 1/2 |
| tan | 1/√3 | 1 | √3 |
Key Point
- sin² θ + cos² θ = 1 (important identity)
- tan θ = sin θ / cos θ
Practice Problems
- State the values of sin 45°, cos 45°, and tan 45°.
- Calculate sin² 60° + cos² 60°.
Solutions and Explanations
- Solution: sin 45° = cos 45° = √2/2, tan 45° = 1.
- Solution: 1. (√3/2)² + (1/2)² = 3/4 + 1/4 = 1.