Section 6.3 – Solving Trigonometric Equations 1

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 2,0 of the equation 1)tan( x .

Example 2: Find all solutions of the equation 1)tan( x .

Example 3: Solve the equation in the interval 2,0 . Then find all solutions to the equation. 22sin 3sin 1x x .

Section 6.3 – Solving Trigonometric Equations 2

Example 4: Solve the equation xxx coscos3cos2 23 in the interval 2,0 .

Example 5: Find all solutions of the equation 20sinsin 2 .

Example 6: Solve the equation cos sin tan 2x x x in the interval 2,0 .

Section 6.3 – Solving Trigonometric Equations 3

Example 7: Solve the equation 01csccot 2 xx in the interval 2,0 .

Example 8: Find all solutions of 1sec2 x in the interval

2

5,

2

.

Section 6.3 – Solving Trigonometric Equations 4

Equations Involving Multiples of an Angle

Example 9: Find all solutions of tan 3 1x in the interval 2,0 .

Example 10: Find all solutions of 12

sin2

x in the interval 2,0 .

Example 11: Find all solutions of 1)9sin( x in the interval

9

2,0

.

Section 6.3 – Solving Trigonometric Equations 5

Example 12: Find all solutions of 1)5tan( x in the interval

10

1,

10

1.

Example 13: Find all solutions in [0, 2 ) for 2

1

32sin

x

Section 6.3 – Solving Trigonometric Equations 6

Example 14: Solve (

) =

√ in [ )

Example 15: Solve ( ) on [

]

Example 16: Solve