Trigonometric Derivatives Overview

Overview

This lecture introduces the derivatives of trigonometric functions (sine, cosine, tangent, etc.) through graphical reasoning and derivative rules, emphasizing key formulas for scientific applications.

Graphical Derivation of Trigonometric Derivatives

• The sine graph reaches its maximum at 1 and minimum at -1 with zeros at 0, π, 2π, etc.
• At the peaks and valleys of sine, the derivative (slope) is 0.
• Where the sine function increases, its derivative is positive; where it decreases, its derivative is negative.
• The slope at certain key points (e.g., at x=0) is exactly 1 if using radians.
• The derivative of sine resembles the shape of the cosine function.

Key Trigonometric Derivative Rules

• The derivative of sin(x) is cos(x).
• The derivative of cos(x) is -sin(x).
• The derivative of tan(x) is sec²(x).
• The derivative of sec(x) is sec(x)tan(x).
• The derivative of csc(x) is -csc(x)cot(x).
• The derivative of cot(x) is -csc²(x).
• Functions containing a "c" (cos, cosec, cot) in their name have a negative sign in their derivative.

Proof Example: Derivative of Secant

• Rewrite sec(x) as 1/cos(x) or (cos(x))⁻¹ for differentiation.
• Apply the chain rule: derivative is (cos(x))⁻² × derivative of cos(x).
• Since derivative of cos(x) is -sin(x), combine and simplify to positive sin(x)/cos²(x).
• Further simplification gives sec(x)tan(x).

Example Problem: Chain Rule with Trig Functions

• Given f(x) = 4 + 6cos(πx² + 1), derivative is:
• Derivative of constant is 0; 6 remains as a multiplier.
• Use chain rule: derivative of cos(…) is -sin(…) times the derivative of the inside (2πx).
• The final derivative: -12πx sin(πx² + 1).

Key Terms & Definitions

• Derivative — Measures the rate of change of a function.
• Sine (sin) — A periodic trig function; its derivative is cosine.
• Cosine (cos) — A periodic trig function; its derivative is negative sine.
• Tangent (tan) — sin(x)/cos(x); derivative is sec²(x).
• Secant (sec) — 1/cos(x); derivative is sec(x)tan(x).
• Chain Rule — Differentiation rule for composite functions.

Action Items / Next Steps

• Practice identifying derivatives of trigonometric functions using chain and quotient rules.
• Review and memorize key trigonometric derivative formulas.
• Complete any assigned problems involving differentiation of trig functions.