Section 6.3
Solving Trigonometric Equations
An equation that contains a trigonometric expression is called a trigonometric
equation. As we know trigonometric functions repeat their behavior. What if
we wanted answers to questions like: “When will the moon look exactly like it
did last night at 9:30pm?” and “When will my breathing be exactly as it is
right now?” The questions to these models can be answered by solving
trigonometric equations.
Example 1: Find all solutions in the interval [0,2π ) of the equation
tan( x ) = −1 .
Example 2: Find all solutions of the equation tan( x ) = −1 .
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Section 6.3 – Solving Trigonometric Equations
1
Example 3: Solve the equation in the interval [0,2π ) . Then find all solutions
−
=−
to the equation.
Example 4: Solve the equation 2 cos 3 x = −3 cos 2 x − cos x in the interval
[0,2π ) .
Section 6.3 – Solving Trigonometric Equations
2
Example 5: Find all solutions of the equation sin 2 θ + sin θ = 20 .
Example 6: Solve the equation osx + sinxtanx = 2 in the interval [0,2π ) .
Example 7: Solve the equation cot 2 x + csc x − 1 = 0 in the interval [0,2π ) .
Section 6.3 – Solving Trigonometric Equations
3
 π 5π 
Example 8: Find all solutions of sec 2 x = 1 in the interval  − ,  .
 2 2
Equations Involving Multiples of an Angle
Example 9: Find all solutions of an 3x = −1 in the interval [0,2π ) .
Section 6.3 – Solving Trigonometric Equations
4
x
Example 10: Find all solutions of 2 sin  = −1 in the interval [0,2π ) .
 2
 2π 
Example 11: Find all solutions of sin( 9x ) = 1 in the interval 0,  .
 9
Section 6.3 – Solving Trigonometric Equations
5
 1 1
Example 12: Find all solutions of − tan( 5πx ) = 1 in the interval  − ,  .
 10 10 
x π 1
Example 13: Find all solutions in [0, 2 π ) for sin −  =
2 3 2
Section 6.3 – Solving Trigonometric Equations
6
Example 14: Solve " # −
$%
&
'=−
√)
Section 6.3 – Solving Trigonometric Equations
in *+, % .
7