AAT Unit 5 Graphing Inverse Trig Functions, Trig Equations and Harmonic Motion. Name: Block: Section Topic Assignment Date Due: 5-7 AAT- 16,19

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1 1 AAT Unit 5 Graphing Inverse Trig Functions, Trig Equations and Harmonic Motion Name: Block: Section Topic Assignment Date Due: 5-7 AAT- 16,19 Inverse Trig Functions Page 566:10-18 even, even Page 5 of packet Quiz on Inverse Trig 6-5 AAT-0 Trig Equations Pages 6-7 of packet Page 9 of packet Page 10 of packet Quiz on Trig Equations 5-8 AAT-9 Applications of Trig Functions Harmonic Motion Page 14 of packet Sunrise time Lab Harmonic Motion Project

2 5.7 Inverse Trig Functions Warm-Up/Review: Fill in the blanks: Unit Circle for Inverse Trig Functions 0

3 3 Inverses: Sin x is sin 1 x or arcsin x Tan x is tan 1 x or arctan x Sec x is sec 1 x or arcsec x Cos x is cos 1 x or arccos x Csc x is csc 1 x or arccsc x Cot x is cot 1 x or arccot x Remember: sin 1 x is NOT csc x! Inverses are used to find angles. Exact Values of Inverse Functions If you are evaluating a positive number: exact solution is in Quadrant I If you are evaluating a negative number: exact solution for sin 1 is in Quadrant IV cos 1 is in Quadrant II tan 1 is in Quadrant IV Examples Evaluate the following inverse trig functions: You Try!

4 4 Composition of Functions and their Inverses: If 1 1 and, then If 1 1 and 0, then If x is any real number and, then Examples: Simplify. 1. arctan You Try: 1. arcsin Using Right Triangles to Evaluate Composition of Functions: Examples: 1. arccos. 3. You Try: 1. arcsin. 3.

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6 6 AAT Review and Practice Solving by Factoring, Unit Circle Solving by Factoring: 1. Set equation equal to zero.. Factor. 3. Set each factor equal to zero and solve. 4. Can have two solutions, one solution, or no solutions. Example 1 m 3m 8 Example 16x 49 Example 3 3x x Solve each equation by factoring. 1. x 4x 1 0. y 16y n 5 10n 4. 9z 10 z 5. 7y 4 y 6. c c w 35w d 4d 45 0

7 7 1. Find the exact value ofcsc Give the exact value of cot Find the exact value of tan Give the exact value ofsec Find the exact value ofsec Find the exact value of tan Find the exact value ofcos Find the exact value ofcsc Find the exact value ofcot Find the exact value ofcot Find the exact value ofsin Find the exact value ofsec Give the exact value ofsin Find the exact value of tan Give the exact value of tan Find the exact value ofsin Give the exact value of cos Give the exact value of csc315.

8 8 AAT Notes 6-5 Solving Trig Equations Warm-up: Evaluate the following. 1. cos 135. sin ( 60 ) 3. tan 5 4. Solve: x x6 0 Trig Identities, like the one to the right, are true for ALL angles: sin x cos x 1 Trig Equations, like the one to the right, are only true for SOME angles: sinx 1 When we solve trig equations, we must find the values of x that make the statement true. The equation, sinx 1, is true when x 90,450,810,... Method to Solve Trig Equations: 1. Get trig function by itself. If you have more than one trig function, solve by factoring.. Use the unit circle to find the angles that make the equation true. 3. Express answers in degrees between 0 and 360. Example 1: Solve sinx 1=0. Example : Solve sinx + = sinx Example 3: Solve3tan x 1 0 Example 4: Solve cos3x 1=0 Example 5: Solve sin x sinx 1 0 Example 6: Solve cot cos x x cotx

9 9 You Try: Solve each equation for 0 x sin x 3 0. cosx cos x sinx 1 4. sin x 3sinx 0 5. cos x cosx sin x cos x 0 7. cosx cosxsinx 0 8. cosx 3cosx

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11 11 AAT Notes-Harmonic Motion Harmonic Motion describes the motion of a point on an object that vibrates, oscillates, rotates, or is moved by wave motion. A traditional example is a mass attached to the end of a spring which bobs up and down. The mass on the spring moves with harmonic motion. Other examples include: vibrations of a guitar string, the pistons of an engine and a pendulum swinging back and forth. Sine and Cosine functions represent harmonic motions very well. When writing harmonic motion equations, try to choose the curve which best represents the given data without causing a horizontal shift. If the motion described begins at a low point or high point, use the cosine function. If the motion begins at the equilibrium, use the sine function. Use the equations: d Acos( Bt C) d Asin( Bt C) where d is the distance from the origin at time t. Some definitions related to harmonic motion: Amplitude an object s maximum displacement from the zero point. Amplitude is A. Period the time it takes to complete one cycle. Period is 360. B B B B Frequency the number of cycles per unit of time. It is the reciprocal of the period 360 Example 1: Given the equation d 6cos135t, find the following: a.) The maximum displacement from zero. b.) The frequency. c.) The value of d when t = 8.

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13 13 Example 4: A weight on a spring bounces a maximum of 8 inches above and below its equilibrium (zero) point. The time for one complete cycle is seconds. Write an equation to describe the motion of this weight. Assume the weight is at equilibrium when t = Sketch a sine curve that illustrates the given information. d 0 t. Use the model d Asin( Bt C) to write the equation: Try these! 1. Write an equation to represent harmonic motion when the initial position is d = -10, the amplitude is 10 and the period is 1.. Given d 4cos8t, find the maximum displacement, the period and the frequency. 3. A ball on a spring is pulled 6 inches below its rest position and then released. The period for the motion is 4 seconds. Write the equation for the ball s harmonic motion.

14 14 AAT PWS 5.8 Simple Harmonic Motion Find the amplitude, period, frequency and phase shift d 3sin t. d 1 cos t 3. 3 d sin t 4. d sint 6 5. d 0.5cos t d 1cos t 6 7. d 0cos1t 8. 3 d 0.3sin t 4 4 Write an equation with phase shift 0 to represent simple harmonic motion under each set of circumstances. 9. initial position 1, amplitude 1, period initial position 0, amplitude period initial position -4, amplitude 4, period 6