Chapter 6
6.5 Trigonometric equations
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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
Dr .Hayk Melikyan
Departmen of Mathematics and CS
melikyan@nccu.edu
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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.
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Equations Involving a Single Trigonometric
Function
•
To solve an equation containing a single trigonometric function:
• Isolate the function on one side of the equation.
sinx = a (-1 ≤ a ≤ 1 )
cosx = a
(-1 ≤ a ≤ 1 )
tan x = a ( for any real a )
• Solve for the variable.
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Trigonometric Equations
y
y = cos x
1
y = 0.5
–4 
–2 
2
4
x
–1
cos x = 0.5 has infinitely many solutions for –  < x < 
y
y = cos x
1
0.5
2
–1
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x
cos x = 0.5 has two solutions for 0 < x < 2
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Text Example
Solve the equation: 3 sin x - 2 = 5 sin x - 1.
Solution
The equation contains a single trigonometric function, sin x.
Step 1 Isolate the function on one side of the equation. We can solve for
sin x by collecting all terms with sin x on the left side, and all the constant
terms on the right side.
3 sin x - 2 = 5 sin x - 1
3 sin x - 5 sin x - 2 = 5 sin x - 5 sin x – 1
-2 sin x - 2 = -1
-2 sin x = 1
sin x = -1/2
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This is the given equation.
Subtract 5 sin x from both sides.
Simplify.
Add 2 to both sides.
Divide both sides by -2 and solve for sin x.
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Text Example
Solve the equation:
2 cos2 x + cos x - 1 = 0,
0  x < 2.
Solution
The given equation is in quadratic form 2t2 + t - 1 = 0 with t =
cos x. Let us attempt to solve the equation using factoring.
2 cos2 x + cos x - 1 = 0
This is the given equation.
(2 cos x - 1)(cos x + 1) = 0
2 cos x - 1= 0
or
Factor. Notice that 2t2 + t – 1 factors as (t – 1)(2t + 1).
cos x + 1 = 0
2 cos x = 1 cos x = -1 cos x = 1/2
Set each factor equal to 0.
Solve for cos x.
x =  x = 2 -=  x = 
The solutions in the interval [0, 2) are /3, , and 5/3.
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Example

Solve the following equation:
7 cos + 9 = -2 cos
Solution:
7 cos  + 9 = -2 cos 
9 cos  = -9
cos  = -1
 =  ,3 ,5
 =  + 2 n
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Example

Solve the equation on the interval [0,2)

3
tan =
2
3
Solution:

3
tan =
2
3

7
= and
2 6
6

7
 = and
3
3
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
8
Example
Solve the equation on the interval [0,2)
cos x + 2 cos x - 3 = 0
2
Solution:
cos 2 x + 2 cos x - 3 = 0
(cos x + 3)(cos x - 1) = 0
cos x + 3 = 0 cos x - 1 = 0
cos x = -3 cos x = 1
no solution
x=0
x=0
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Example

Solve the equation on the interval [0,2)
sin 2x = sin x
Solution:
sin 2 x = sin x
2 sin x cos x = sin x
2 cos x = 1
1
cos x =
2

5
x= ,
3
3
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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
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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
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Solving an Equation with a Multiple Angle

Solve the equation:
tan

3
tan 2 x = 3,0  x < 2 .
= 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
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Solving an Equation with a Multiple Angle

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
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Solving a Trigonometric Equation Quadratic in Form

Solve the equation:
4cos 2 x - 3 = 0
4cos x = 3
2
3
cos x =
4
4cos 2 x - 3 = 0, 0  x < 2 .
The solutions in the interval  0,2 
for this equation are:
2
3
3
cos x = 
=
4
2
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 5 7
11
, , , and
.
6 6 6
6
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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
x =0 x =
tan x = 1
x=

4
5
x=
4
The solutions for this equation in the interval  0,2  are:

5
0, ,  , and
.
4
4
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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
7
11
2sin x + 1 = 0
 , , and
.
6
6
2sin x = -1
sin x - 1 = 0
1
sin x = sin x = 1
2
x =
7
11
x=
x=
6
6

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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.
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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.
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