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3.5 Derivatives of Trigonometric Functions Quick Review 1. Convert 135 degrees to radian. 3 2.356 4 2. Convert 1.7 radian to degrees. 97.403o 3. Find the exact value of sin without a calculator. 3 4. State the domain and range of the cosine function. Domain : all reals range : 1, 1 5. State the domain and range of the tangent function. k k odd number Domain : x 2 range : all reals 3 2 Quick Review 6. If sin a = – 1, what is cos a? 0 7. If tan a = – 1, what are two possible values of sin a? 2 2 2 1 cos h sin h 8. Verify the identity . h h1 cos h 9. Find an equation of the line tangent to the curve y 2 x3 7 x 2 10 at the point (3, 1). y 12 x 35 10.A particle moves along a line with velocity v 2t 3 7t 2 10 for time t > 0. Find the acceleration of the particle at t = 3. 12 What you’ll learn about Derivative of the Sine Function Derivative of the Cosine Function Simple Harmonic Motion Jerk Derivatives of Other Basic Trigonometric Functions Essential Questions What are the derivatives of sines and cosines, and how do they play a key role in describing periodic change? Finding the Derivative of the Sine Function 1. Use the definition of derivative to find the derivative of f (x) = sin x. sin x h sin x lim h 0 h sin x cos h cos x sin h sin x lim h 0 h sin xcos h 1 cos x sin h lim lim h 0 h 0 h h sin h cos h 1 lim cos x lim sin x h 0 h 0 h h 0 cos x cos x sin x lim 1 x 0 x cos x 1 lim 0 x 0 x Derivative of the Sine Function The derivative of the sine is the cosine. d sin x cos x dx Derivative of the Cosine Function The derivative of the cosine is the negative of the sine. d cos x sin x dx Using the Derivative of Sine and Cosine 3. Find the derivative of each: a) y x sin x 2 y x cos x sin x 2 x 2 2 y x cos x 2 x sin x cos x b) u 1 sin x 1 sin x sin x cos x cos x u 2 1 sin x 1 sin x sin x sin 2 x cos 2 x 2 2 1 sin x 1 sin x 1 1 sin x Simple Harmonic Motion The motion of a weight bobbing up and down on the end of a string is an example of simple harmonic motion. Example Simple Harmonic Motion 4. A weight hanging from a spring is stretched 5 units beyond its rest position (s = 0) and released at time t = 0 to bob up and down. Its position at any later time t is s 5 cos t What are its velocity and acceleration at time t? vt 5 sin t at 5 cos t s 5 sin t cos t 0 Jerk Jerk is the derivative of acceleration. If a body's position at time t is da d 3 s j t 3. dt dt 5. Find the jerk caused by the constant acceleration of gravity (g = – 32 ft/sec 2) j t 0 6. Find the jerk of the simple harmonic motion of the previous example s = 5 cos t. vt 5 sin t at 5 cos t j t 5 sin t Finding the Derivative of the Tan Function 7. Use the derivative of sin and cos to find the derivative of f (x) = tan x. sin x f x tan x cos x cos x cos x sin x sin x f x 2 cos x 2 2 1 cos x sin x 2 2 cos x cos x 2 sec x Derivative of the Other Basic Trig Functions d tan x sec 2 x dx d cot x csc 2 x dx d sec x sec x tan x dx d csc x csc x cot x dx Example Derivative of the Other Basic Trigonometric Functions dy cot x 8. Find for y . dx 1 cot x dy 1 cot x csc x cot x csc 2 x 2 dx 1 cot x csc 2 x cot x csc 2 x cot x csc 2 x 2 1 cot x csc x 2 1 cot x 2 2 Example Finding Horizontal Tangents of Trigonometric Functions 9. Show that the graphs of y = sec x and y = cos x have horizontal tangents at x = 0. y sec x y sec x tan x 0 sec x 0 tan x 0 x0 y cos x y sin x 0 sin x 0 x0 Example Derivative of Trig Functions 10. Find y if y sec x. y sec x tan x 2 y sec x sec x tan x sec x tan x sec x tan x sec x 3 2 Example Derivative of Trig Functions 11. Find y if y cos . y sin cos 1 sin cos y cos sin 1 sin cos 2 sin Pg. 146, 3.5 #2-48 even