Chapter 5 Analytic Trigonometry Copyright © 2014, 2010, 2007 Pearson Education, Inc. 1 5.5...
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Transcript of Chapter 5 Analytic Trigonometry Copyright © 2014, 2010, 2007 Pearson Education, Inc. 1 5.5...
Chapter 5AnalyticTrigonometry
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 1
5.5 TrigonometricEquations
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 2
Objectives:
• Find all solutions of a trigonometric equation.• Solve equations with multiple angles.• Solve trigonometric equations quadratic in form.• Use factoring to separate different functions in
trigonometric equations.• Use identities to solve trigonometric equations.• Use a calculator to solve trigonometric equations.
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 3
Trigonometric Equations and Their Solutions
A trigonometric equation is an equation that contains a trigonometric expression with a variable, such as sin x.
The values that satisfy such an equation are its solutions. (There are trigonometric equations that have no solution.)
When an equation includes multiple angles, the period of the function plays an important role in ensuring that we do not leave out any solutions.
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 4
Example: Finding all Solutions of a Trigonometric Equation
Solve the equation:
Step 1 Isolate the function on one side of the equation.
5sin 3sin 3. x x
5sin 3sin 3x x
5sin 3sin 3sin 3sin 3 x x x x
2sin 3x
3sin
2x
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 5
Example: Finding all Solutions of a Trigonometric Equation (continued)
Solve the equation:
Step 2 Solve for the variable.
5sin 3sin 3. x x
3sin
2x
Solutions for this equation in are: 0,2
The solutions for this equation are:
2,
3 3
22 , 2
3 3n n
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 6
Example: Solving an Equation with a Multiple Angle
Solve the equation: tan 2 3,0 2 . x x
tan 33
Because the period is all solutions for this equation are given by
,
23
x n
6 2n
x
0n 06 2 6
x
1n3 4 2
6 2 6 6 6 3x
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 7
Example: Solving an Equation with a Multiple Angle (continued)
Solve the equation: tan 2 3,0 2 . x x
Because the period is all solutions for this equation are
given by
,.
6 2 nx
2n
3n
2 6 76 2 6 6 6
x
3 9 10 56 2 6 6 6 3
x
In the interval , the solutions are:2 7 5
, , , and .6 3 6 3 0,2
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 8
Example: Solving a Trigonometric Equation Quadratic in Form
Solve the equation: 24cos 3 0, 0 2 . x x24cos 3 0x
24cos 3x
2 3cos
4x
3 3cos
4 2x
The solutions in the interval
for this equation are:
5 7 11, , , and .
6 6 6 6
0,2
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 9
Example: Using Factoring to Separate Different Functions
Solve the equation: sin tan sin , 0 2 . x x x xsin tan sinx x xsin tan sin 0x x x
sin (tan 1) 0x x
sin 0x
0 x x
tan 1 0x
tan 1x
5
4 4 x x
The solutions for this equation in the interval are:5
0, , , and .4 4
0,2
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 10
Example: Using an Identity to Solve a Trigonometric Equation
Solve the equation: cos2 sin 0, 0 2 . x x xcos2 sin 0x x
21 2sin sin 0x x 22sin sin 1 0x x
(2sin 1)(sin 1) 0x x 2sin 1 0x
2sin 1x 1
sin2
x
7 11
6 6 x x
sin 1 0x
sin 1x
x
The solutions in the
interval are7 11
, , and .6 6
0,2
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 11
Example: Solving Trigonometric Equations with a Calculator
Solve the equation, correct to four decimal places, for0 2 .x
tan 3.1044x 1tan (3.1044)x
1.2592x
tanx is positive in quadrants I and III
In quadrant I
In quadrant III
1.2592x
1.2592x
4.4008
The solutions for this equation are 1.2592 and 4.4008.
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 12
Example: Using a Calculator to Solve Trigonometric Equations
Solve the equation, correct to four decimal places, for0 2 .x
sin 0.2315x 1sin ( 0.2315)x
0.2336x
Sin x is negative in quadrants III and IV
In quadrant III
In quadrant IV 2 1.2592
6.0496
x
x
The solutions for this equation are 3.3752 and 6.0496.
0.2336
3.3752
x
x