Math 1304 Calculus I 3.2 – Derivatives of Trigonometric Functions

Trigonometric Functions Measure –Radians, degrees Basic functions –sin, cos, tan, csc, sec, cot Periodicity Special values at: –0, π/6, π/4, π/3, π/2, π Sign change Addition formulas Derivatives

Angle Radians: Measure angle by arc length around unit circle θ

Definition of Basic Functions sin(  ) = opposite / hypotenuse cos(  ) = adjacent / hypotenuse tan(  ) = opposite / adjacent csc(  ) = hypotenuse / opposite sec(  ) = hypotenuse / adjacent cot(  ) = adjacent / opposite hypotenuse opposite adjacent θ

Sin and Cos Give the Others

Sin, Cos, Tan on Unit Circle θ θ cos(θ) sin(θ) 1 tan(θ)

Periodicity

Special Values

Basic Inequalities θ θ cos(θ) sin(θ) 1 tan(θ) For

Proof of Basic Equalities θ θ cos(θ) sin(θ) 1 tan(θ) D E A B C Draw tangent line at B. It intersects AD at E O

Special Limit

Use Squeezing Theorem

Another Special Limit

Addition Formulas sin(x+y) = sin(x) cos(y) + cos(x) sin(y) cos(x+y) = cos(x) cos(y) – sin(x) sin(y)

Derivative of Sin and Cos Use addition formulas (in class)

Derivatives If f(x) = sin(x), then f’(x) = cos(x) If f(x) = cos(x), then f’(x) = - sin(x) If f(x) = tan(x), then f’(x) = sec 2 (x) If f(x) = csc(x), then f’(x) = - csc(x) cot(x) If f(x) = sec(x), then f’(x) = sec(x) tan(x) If f(x) = cot(x), then f’(x) = - csc 2 (x)

A good working set of rules Constants: If f(x) = c, then f’(x) = 0 Powers: If f(x) = x n, then f’(x) = nx n-1 Exponentials: If f(x) = a x, then f’(x) = (ln a) a x Trigonometric Functions: If f(x) = sin(x), then f’(x)=cos(x) If f(x) = cos(x), then f’(x) = -sin(x) If f(x)= tan(x), then f’(x) = sec 2 (x) If f(x) = csc(x), then f’(x) = -csc(x) cot(x) If f(x)= sec(x), then f’(x) = sec(x)tan(x) If f(x) = cot(x), then f’(x) = -csc 2 (x) Scalar mult: If f(x) = c g(x), then f’(x) = c g’(x) Sum: If f(x) = g(x) + h(x), then f’(x) = g’(x) + h’(x) Difference: If f(x) = g(x) - h(x), then f’(x) = g’(x) - h’(x) Multiple sums: derivative of sum is sum of derivatives Linear combinations: derivative of linear combo is linear combo of derivatives Product: If f(x) = g(x) h(x), then f’(x) = g’(x) h(x) + g(x)h’(x) Multiple products: If F(x) = f(x) g(x) h(x), then F’(x) = f’(x) g(x) h(x) + … Quotient: If f(x) = g(x)/h(x), then f’(x) = (g’(x) h(x) - g(x)h’(x))/(h(x)) 2