Fix the mistake: Figures and Measurement

Geometry and Measurement ①: Definitions of Trigonometric Ratios

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).

Commonly Used Trigonometric Ratios

θ30°45°60°
sin1/2√2/2√3/2
cos√3/2√2/21/2
tan1/√31√3
Key Point
  • sin² θ + cos² θ = 1 (important identity)
  • tan θ = sin θ / cos θ

Practice Problems

  1. State the values of sin 45°, cos 45°, and tan 45°.
  2. Calculate sin² 60° + cos² 60°.

Solutions and Explanations

  1. Solution: sin 45° = cos 45° = √2/2, tan 45° = 1.
  2. Solution: 1. (√3/2)² + (1/2)² = 3/4 + 1/4 = 1.

Math (Math I): related lessons