Lecture Notes: Basic Trigonometric Functions

Introduction

• Discussion on basic trigonometric functions: sine, cosine, tangent, cotangent, secant, cosecant.
• Unit circle used to illustrate trigonometric functions.
• Coordinates (x, y) on unit circle relate to trigonometric functions: (cosine θ, sine θ).

Trigonometric Functions and Right Triangles

• Right triangle components: hypotenuse (HYP), adjacent (ADJ), opposite (OPP).
• Projections on axes:
  • Sine (θ) = opposite/hypotenuse
  • Cosine (θ) = adjacent/hypotenuse

Tangent and Cotangent

• Tangent (θ) = opposite/adjacent or sine(θ)/cosine(θ)
• Cotangent (θ) = cosine(θ)/sine(θ) (reciprocal of tangent)

Secant and Cosecant

• Secant (θ) = 1/cosine(θ)
• Cosecant (θ) = 1/sine(θ)

Graphs and Domains

• Trigonometric functions are periodic.

Sine Function

• Domain: (-∞, ∞)
• Key points: 0 at 0, π, 2π; 1 at π/2; -1 at 3π/2.

Cosine Function

• Domain: (-∞, ∞)
• Starts at 1.
• Key points: 0 at π/2, 3π/2; -1 at π.

Tangent Function

• Domain: all real numbers except π/2 + nπ, where n is an integer.
• Vertical asymptotes at points where cosine(θ) = 0.

Cotangent Function

• Domain: all real numbers except nπ.
• Vertical asymptotes at points where sine(θ) = 0.

Secant Function

• Domain: all real numbers except π/2 + nπ.

Cosecant Function

• Domain: all real numbers except nπ.

Special Angles and Values

• Important angles: 0, π/6, π/4, π/3, π/2.
• Memorization of sine and cosine values at these angles.
• Relationships and patterns between sine and cosine values.

Sign of Trigonometric Functions by Quadrant

• Quadrant I: All functions are positive.
• Quadrant II: Sine and cosecant positive.
• Quadrant III: Tangent and cotangent positive.
• Quadrant IV: Cosine and secant positive.

Odd and Even Functions

• Even functions: Cosine and secant.
• Odd functions: Sine, tangent, cotangent, cosecant.

Pythagorean Identities

1. sin²(t) + cos²(t) = 1.
2. 1 + tan²(t) = sec²(t).
3. 1 + cot²(t) = csc²(t).

Examples

• Calculate exact values for trigonometric functions at given angles.
• Use of reference numbers and quadrant information to determine signs.
• Examples include tangent of 5π/6, 7π/6, 11π/6, sine and cosine of 2π/3.

Conclusion

• Importance of memorizing formulas, values, and their signs.
• Encouragement to pause and work through problems independently for better understanding.