Fix the mistake: Trigonometric Functions ① (Basic Definitions, the Unit Circle, and Radians)

Radian Measure and Angle Conversion

To work with trigonometric functions, angles must be expressed in radians (radian measure). Convert them based on 180° = π rad。

Conversion Formulas

  • Degrees → radians: × π/180
  • Radians → degrees: × 180/π

Common angles: 30°=π/6、45°=π/4、60°=π/3、90°=π/2、120°=2π/3、180°=π、270°=3π/2、360°=2π

Example ①
Express the following in radians: (1) 60°　(2) 135°　(3) 270°
Solution:(1) π/3　(2) 3π/4　(3) 3π/2
Example ②
Find the arc length and area of a sector with radius 6 and central angle 2π/3.
Solution:l = 6 × 2π/3 = 4π、S = (1/2) × 36 × 2π/3 = 12π
Key Points
  • 180° = π rad / Arc length l = rθ / Area S = r²θ/2

Practice Problems

  1. Express the following in radians: (1) 120°　(2) 315°　(3) 30°
  2. Express the following in degrees: (1) 7π/6　(2) π/4　(3) 4π/3
  3. Find the arc length and area of a sector with radius 4 and central angle 3π/4.

Solutions and Explanations

  1. Answer:(1) 2π/3　(2) 7π/4　(3) π/6
  2. Answer:(1) 210°　(2) 45°　(3) 240°
  3. Answer:Arc length 3π, area 6π
    Explanation: l=4×3π/4=3π、S=(1/2)×16×3π/4=6π。

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