6 Inverse Circular Functions and Trigonometric Equations Copyright

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6 Inverse Circular Functions and Trigonometric Equations Copyright © 2017, 2013, 2009 Pearson Education,

6 Inverse Circular Functions and Trigonometric Equations Copyright © 2017, 2013, 2009 Pearson Education, Inc. 1

6. 3 Trigonometric Equations II ▪ Equations with Half-Angles ▪ Equations with Multiple Angles

6. 3 Trigonometric Equations II ▪ Equations with Half-Angles ▪ Equations with Multiple Angles ▪ Applications Copyright © 2017, 2013, 2009 Pearson Education, Inc. 2

Example 1 SOLVING AN EQUATION WITH A HALFANGLE (a) over the interval and (b)

Example 1 SOLVING AN EQUATION WITH A HALFANGLE (a) over the interval and (b) for all solutions. The two numbers over the interval value Copyright © 2017, 2013, 2009 Pearson Education, Inc. with sine 3

Example 1 SOLVING AN EQUATION USING A HALF -ANGLE IDENTITY (continued) This is a

Example 1 SOLVING AN EQUATION USING A HALF -ANGLE IDENTITY (continued) This is a sine curve with period The x-intercepts are the solutions found in Example 1(a). Using Xscl = makes it possible to support the exact solutions by counting the tick marks from 0 on the graph. Copyright © 2017, 2013, 2009 Pearson Education, Inc. 4

Example 2 SOLVING AN EQUATION USING A DOUBLE ANGLE IDENTITY Factor. or Copyright ©

Example 2 SOLVING AN EQUATION USING A DOUBLE ANGLE IDENTITY Factor. or Copyright © 2017, 2013, 2009 Pearson Education, Inc. 5

Example 3 a SOLVING AN EQUATION USING A MULTIPLE-ANGLE IDENTITY From the given interval

Example 3 a SOLVING AN EQUATION USING A MULTIPLE-ANGLE IDENTITY From the given interval 0° ≤ θ < 360°, the interval for 2θ is 0° ≤ 2θ < 720°. Solution set: {30°, 60°, 210°, 240°} Copyright © 2017, 2013, 2009 Pearson Education, Inc. 6

Example 3 b Solve SOLVING AN EQUATION USING A MULTIPLE-ANGLE IDENTITY for all solutions.

Example 3 b Solve SOLVING AN EQUATION USING A MULTIPLE-ANGLE IDENTITY for all solutions. All angles 2θ that are solutions of the equation are found by adding integer multiples of 360° to the basic solution angles, 60° and 120°. Solution set, where 180º represents the period of sin 2θ: {30° + 180°n, 60° + 180°n, where n is any integer} Copyright © 2017, 2013, 2009 Pearson Education, Inc. 7

Example 4 SOLVING AN EQUATION WITH A MULTIPLE ANGLE Solve tan 3 x +

Example 4 SOLVING AN EQUATION WITH A MULTIPLE ANGLE Solve tan 3 x + sec 3 x = 2 over the interval One way to begin is to express everything in terms of secant. Square both sides. Copyright © 2017, 2013, 2009 Pearson Education, Inc. 8

Example 4 SOLVING AN EQUATION WITH A MULTIPLE ANGLE (continued) Multiply each term of

Example 4 SOLVING AN EQUATION WITH A MULTIPLE ANGLE (continued) Multiply each term of the inequality find the interval for 3 x: by 3 to Using a calculator and the fact that cosine is positive in quadrants I and IV, we have Copyright © 2017, 2013, 2009 Pearson Education, Inc. 9

Example 4 SOLVING AN EQUATION WITH A MULTIPLE ANGLE (continued) Since the solution was

Example 4 SOLVING AN EQUATION WITH A MULTIPLE ANGLE (continued) Since the solution was found by squaring both sides of an equation, we must check that each proposed solution is a solution of the original equation. Solution set: {0. 2145, 2. 3089, 4. 4033} Copyright © 2017, 2013, 2009 Pearson Education, Inc. 10

Frequencies of Piano Keys A piano string can vibrate at more than one frequency.

Frequencies of Piano Keys A piano string can vibrate at more than one frequency. It produces a complex wave that can mathematically be modeled by a sum of several pure tones. When a piano key with a frequency of f 1 is played, the corresponding string vibrates not only at f 1 but also at 2 f 1, 3 f 1, 4 f 1, …, nf 1. f 1 is the fundamental frequency of the string, and higher frequencies are the upper harmonics. The human ear will hear the sum of these frequencies as one complex tone. (Source: Roederer, J. , Introduction to the Physics and Psychophysics of Music, Second Edition, Springer-Verlag. ) Copyright © 2013, 2009, 2005 Pearson Education, Inc. 11

Example 5 ANALYZING PRESSURES OF UPPER HARMONICS Suppose that the A key above middle

Example 5 ANALYZING PRESSURES OF UPPER HARMONICS Suppose that the A key above middle C is played on a piano. Its fundamental frequency is f 1 = 440 Hz and its associate pressure is expressed as The string will also vibrate at f 2 = 880, f 3 = 1320, f 4 = 1760, f 5 = 2200, … Hz. The corresponding pressures are Copyright © 2017, 2013, 2009 Pearson Education, Inc. 12

Example 5 ANALYZING PRESSURES OF UPPER HARMONICS (continued) The graph of P = P

Example 5 ANALYZING PRESSURES OF UPPER HARMONICS (continued) The graph of P = P 1 + P 2 + P 3 + P 4 + P 5 is “saw-toothed. ” (a) What is the maximum value of P? (b) At what values of t = x does this maximum occur over the interval [0, 0. 01]? Copyright © 2017, 2013, 2009 Pearson Education, Inc. 13

Example 5 ANALYZING PRESSURES OF UPPER HARMONICS (continued) A graphing calculator shows that the

Example 5 ANALYZING PRESSURES OF UPPER HARMONICS (continued) A graphing calculator shows that the maximum value of P is approximately 0. 00317. The maximum occurs at t = x ≈ 0. 000191, 0. 00246, 0. 00474, 0. 00701, and 0. 00928. Copyright © 2017, 2013, 2009 Pearson Education, Inc. 14