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```Chapter 5
Analytic
Trigonometry
5.5 Trigonometric
Equations
1
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.
2
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.
3
Example: Finding all Solutions of a Trigonometric
Equation
Solve the equation: 5sin x  3sin x  3.
Step 1 Isolate the function on one side of the equation.
5sin x  3sin x  3
5sin x  3sin x  3sin x  3sin x  3
2sin x  3
3
sin x 
2
4
Example: Finding all Solutions of a Trigonometric
Equation
(continued)
Solve the equation: 5sin x  3sin x  3.
Step 2 Solve for the variable.
3
sin x 
2
Solutions for this equation in  0,2  are:
 2
,
3 3

2
The solutions for this equation are:  2n ,  2n
3
3
5
Example: Solving an Equation with a Multiple Angle
Solve the equation: tan 2 x  3,0  x  2 .
6
Example: Solving an Equation with a Multiple Angle
Solve the equation: tan 2 x  3,0  x  2 .
tan

3
 3
Because the period is  , all solutions for
this equation are given by

0



n0
x 

2 x   n
6 2 6
3
 n
x 
   3 4 2


n 1 x    
6 2
6 2 6 6
6
3
7
Example: Solving an Equation with a Multiple Angle
(continued)
Solve the equation: tan 2 x  3,0  x  2 .
Because the period is  , all solutions for this equation are
 n
given by x   .
6 2
n2
n3

2  6 7
x 
 

6 2 6 6
6
 3  9 10 5
x 
 


6 2 6 6
6
3

2 7
5
0,2

 , the solutions are: , , , and .
In the interval 
6 3 6
3
8
Example: Solving a Trigonometric Equation Quadratic in
Form
Solve the equation: 4cos x  3  0, 0  x  2 .
2
4cos 2 x  3  0
The solutions in the interval  0,2 
4cos x  3
2
for this equation are:
3
2
cos x 
4
 5 7
3
3
cos x  

4
2
11
, , , and
.
6 6 6
6
9
Example: Using Factoring to Separate Different
Functions
Solve the equation: sin x tan x  sin x, 0  x  2 .
sin x tan x  sin x
sin x tan x  sin x  0
tan x  1  0
sin x(tan x  1)  0
sin x  0
tan x  1
x

x 0 x 
4
5
x
4
The solutions for this equation in the interval  0,2  are:

5
0, ,  , and
.
4
4
10
Example: Using an Identity to Solve a Trigonometric
Equation
Solve the equation: cos2 x  sin x  0, 0  x  2 .
cos 2 x  sin x  0
1  2sin 2 x  sin x  0
The solutions in the
2sin 2 x  sin x  1  0
interval  0,2  are
(2sin x  1)(sin x  1)  0
p
7p
11p
2sin x  1  0
, , and
.
2 6
6
2sin x  1
sin x  1  0
1
sin x  
sin x  1
2
p
x=
7
11
x
x
2
6
6
11
Example: Solving Trigonometric Equations with a
Calculator
Solve the equation, correct to four decimal places, for
0  x  2 .
tanx is positive in quadrants I and III
tan x  3.1044
In quadrant I x  1.2592
1
x  tan (3.1044)
x  1.2592
In quadrant III x    1.2592
 4.4008
The solutions for this equation are 1.2592 and 4.4008.
12
Example: Using a Calculator to Solve Trigonometric
Equations
Solve the equation, correct to four decimal places, for
0  x  2 .
Sin x is negative in quadrants III and IV
sin x  0.2315
In quadrant III x    0.2336
1
x  sin (0.2315)
x  3.3752
x  0.2336
In quadrant IV x  2  1.2592
x  6.0496
The solutions for this equation are 3.3752 and 6.0496.