Trigonometric Functions
3.1 Introduction
Definition: Trigonometric functions relate angles to the ratios of sides in a right triangle.
Applications: Used in geometry, physics, engineering, and periodic phenomena (waves, oscillations).
Key Concepts:
- Unit circle approach.
- Angle measurement (degrees, radians).
3.2 Angles
Types of Angles:
- Acute (<90°), Right (90°), Obtuse (>90°), Straight (180°).
Units:
- Degrees (°) and Radians (1 rad ≈ 57.3°).
- Conversion: π rad = 180°.
Angle Conventions:
- Positive (counter-clockwise), Negative (clockwise).
- Coterminal angles (same terminal side, e.g., 30° and 390°).
3.3 Trigonometric Functions
Primary Functions: Defined for an angle θ in a right triangle or unit circle.
- Sine (sinθ): Opposite/Hypotenuse (y/r).
- Cosine (cosθ): Adjacent/Hypotenuse (x/r).
- Tangent (tanθ): Opposite/Adjacent (y/x).
Reciprocal Functions:
- Cosecant (cscθ = 1/sinθ), Secant (secθ = 1/cosθ), Cotangent (cotθ = 1/tanθ).
Unit Circle:
- For any angle θ, (x, y) = (cosθ, sinθ).
Signs in Quadrants:
| Quadrant | Positive Functions |
|---|---|
| I | All |
| II | sin, csc |
| III | tan, cot |
| IV | cos, sec |
3.4 Trigonometric Functions of Sum and Difference of Two Angles
Key Identities:
- sin(A ± B) = sinA cosB ± cosA sinB
- cos(A ± B) = cosA cosB ∓ sinA sinB
- tan(A ± B) = (tanA ± tanB)/(1 ∓ tanA tanB)
Applications:
- Simplifying expressions.
- Solving trigonometric equations.
- Deriving double-angle/half-angle formulas.
Example:
Find sin(15°) using 15° = 45° - 30°:
sin(45° - 30°) = sin45° cos30° - cos45° sin30° = (√2/2)(√3/2) - (√2/2)(1/2) = (√6 - √2)/4
Visual Summary:
- Unit Circle: Memorize key angles (0°, 30°, 45°, 60°, 90°) and their sine/cosine values.
- Graphs: sinθ and cosθ are periodic (2π), tanθ has period π.