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Chapter 3.5 Derivatives of Trigonometric Functions Objectives β’ Use the rules for differentiating the six basic trigonometric functions. Learning Target β’ 80% of the students will be able to calculate the derivative of csc π₯. Standard G-C.4 Construct a tangent line from a point outside a given circle to the circle. Overview β’ β’ β’ β’ β’ Derivative of the Sine Function Derivative of the Cosine Function Simple Harmonic Motion Jerk Derivatives of Other Basic Trigonometric Functions Periodic Motion β’ Many phenomena are periodic. β’ Sound β’ Light β’ Ocean waves β’ Tides β’ Tsunamis β’ Earthquakes β’ Heart rates Derivative of sin π₯ π¬π’π§π π /π π π¬π’π§π π¬π’π§ π π cos π β π π 1 0 -1 Derivative of sin π₯ Let π¦ = sin π₯ ππ¦ sin π₯ + β β sin π₯ = lim ππ₯ ββ0 β sin π₯ cos β + cos π₯ sin β β sin π₯ = lim ββ0 β sin π₯ cos β β 1 + cos π₯ sin β = lim ββ0 β cos β β 1 sin β = lim sin π₯ β lim + lim cos π₯ β lim ββ0 ββ0 ββ0 ββ0 β β = sin π₯ β 0 + cos π₯ β 1 = cos π₯ Derivative of sin π₯ and cos π₯ π sin π₯ = cos π₯ ππ₯ π cos π₯ = β sin π₯ ππ₯ The sine and cosine functions are differentiable. Continuous over their domains. Also true of the other basic trigonometric functions Using Degrees Example 1 a) Find the derivative of π¦ = π₯ 2 sin π₯. ππ¦ π π 2 2 =π₯ β sin π₯ + sin π₯ π₯ ππ₯ ππ₯ ππ₯ = π₯ 2 cos π₯ + 2π₯ sin π₯ b) Find the derivative of π’ = cos π₯ 1βsin π₯ π π 1 β sin π₯ cos π₯ β cos π₯ 1 β sin π₯ ππ’ ππ₯ ππ₯ = ππ₯ 1 β sin π₯ 2 1 β sin π₯ β sin π₯ β cos π₯ 0 β cos π₯ = 1 β sin π₯ 2 β sin π₯ + sin2 π₯ + cos 2 π₯ = 1 β sin π₯ 2 1 β sin π₯ 1 = = 2 1 β sin π₯ 1 β sin π₯ Exercise 1 Find the derivative of a) π¦ = 3π₯ + π₯ tan π₯ b) π¦ = cos π₯ 1+sin π₯ Example 2 Simple Harmonic Motion A weight hanging from a spring is stretched 5 units below its rest position (π = 0) and released at π‘ = 0 to bob up and down. Its position at any time t is π = 5 cos π‘. What are its velocity and acceleration at t? Solution β’ Position π = 5 cos π‘ β’ Velocity π π£= 5 cos π‘ = β5 sin π‘ ππ‘ β’ Acceleration ππ£ π π= = β5 sin π‘ = β5 cos π‘ ππ‘ ππ‘ π = π ππ¨π¬ π 6 5 4 3 2 1 0 -1 -2 -3 -4 -5 -6 π = βπ π¬π’π§ π π = βπ ππ¨π¬ π Exercise 2 A body is moving in simple harmonic motion with position π = cos π‘ β 3 sin π‘. 1. Find its velocity, speed, and acceleration at time t. 2. Find its velocity, speed, and acceleration at time π‘ = π 4. Jerk β’ Not the boyfriend who dumped you. β’ Sudden change in acceleration. β’ The 3rd derivative of position. Definition Jerk is the derivative of acceleration. If a bodyβs position at time t is given by π π‘ , ππ π 3 π π π‘ = = 3. ππ‘ ππ‘ Example 3 1. The jerk caused by the acceleration of gravity (π = 9.8 π π 2 ) is zero. π π= π=0 ππ‘ We do not experience motion sickness while sitting in a chair. Example 3 2. The jerk of the simple harmonic motion of Example 2 is, ππ π π= = β5 cos π‘ ππ‘ ππ‘ = 5 sin π‘ Greatest magnitude when body at equilibrium position (sin π‘ = ±1) where acceleration is zero. Exercise 3 A body is moving in simple harmonic motion with its position given by, π π‘ = 2 + 2 sin π‘ Find the jerk at time t. Other Trigonometric Functions sin π₯ tan π₯ = cos π₯ π tan π₯ = sec 2 π₯ ππ₯ 1 csc π₯ = sin π₯ π csc π₯ = β csc π₯ cot π₯ ππ₯ 1 sec π₯ = cos π₯ cos π₯ cot π₯ = sin π₯ π sec π₯ = sec π₯ tan π₯ ππ₯ π cot π₯ = β csc 2 π₯ ππ₯ Example 4 Find the equations of the lines that are tangent and normal to tan π₯ π π₯ = π₯ at π₯ = 2. support graphically. Exercise 4 Find the equations of the lines that are tangent and normal to the graph of π¦ = sec π₯ at π₯ = π 4. Example 5 Find π¦ β²β² if π¦ = sec π₯. π¦ = sec π₯ π¦ β² = sec π₯ tan π₯ π π¦ β²β² = sec π₯ tan π₯ ππ₯ π π = sec π₯ tan π₯ + tan π₯ sec π₯ ππ₯ ππ₯ = sec π₯ sec 2 π₯ + tan π₯ sec π₯ tan π₯ = sec 3 π₯ + sec π₯ tan2 π₯ Exercise 5 Find π¦ β²β² if π¦ = π tan π. Homework p 146: 3-33 odd, 35, 37, 43