Solving Trigonometric Equations

Sec. 5.1bHW: p. 456 39-67 odd

Let’s start with a problem…

Solve

3coscot

sin

xx

x in the interval 0,2

3cos cos

sin sin

x x

x x

3cos cosx x3cos cos 0x x

2cos cos 1 0x x

2cos sin 0x x

cos 0x or sin 0x Reject sin(x) = 0…why???

3,2 2

x

So where is

0,2cos 0x

in ?

Solving a Trig. Eq. by Factoring

Find all solutions to the equation 22sin sin 1x x Let sinu x

Substitute: 22 1u u 22 1 0u u

2 1 1 0u u 2 1 0u or 1 0u 1

2u or 1u

1sin

2x or sin 1x

Solving a Trig. Eq. by Factoring

Find all solutions to the equation 22sin sin 1x x First, find thesolutions in the interval

1sin

2x or sin 1x 0,2

5,6 6

x

3

2x

To get all real solutions, simply add integer multiples of the period:

26

x n or

52

6x n

or3

22

x n

0, 1, 2,n

Solving a Trig. Eq. with a Calculator

Find all solutions to the equation , using a calculatorwhere needed.

cos 0.7t

0.795t To get all real solutions, simply add integer multiples of the period:

0.795 2t n 0, 1, 2,n

Find the solutions in the interval by graphing: ,

Guided PracticeFind all solutions to the given equation in the interval .Do not use a calculator.

Factor:

2 tan cos tan 0x x x tan 2 cos 1 0x x

0,2

tan 0x or1

cos2

x

0,x 7,4 4

x

Final Answer:7

0, , ,4 4

x

Guided PracticeFind all solutions to the given equation in the interval .Do not use a calculator.

2sin tan sinx x x

0,2

2sin tan sin 0x x x 2sin tan 1 0x x

sin 0x or 2tan 1x

0,x 3 5 7, , ,4 4 4 4

x

Final Answer:3 5 7

0, , , , ,4 4 4 4

x

Guided PracticeFind all solutions to the given equation. Do not use a calculator.

22sin 3sin 1 0x x 2sin 1 sin 1 0x x

1sin

2x 7 3 11

, ,6 2 6

x

Initial answers:

Final Answers:

7 3 112 , 2 , 2

6 2 6x n n n

(n an integer)

or sin 1x

Guided PracticeFind all solutions to the given equation. Do not use a calculator.

23sin 2cost t 23sin 2 2sint t

Final Answers:5

2 , 26 6

t n n

(n an integer)

1sin

2t or sin 2t Can the sine of an

angle ever equal –2???5,6 6

t

Initial answers:

22sin 3sin 2 0t t 2sin 1 sin 2 0t t