Sect. 5-3

What is SOLVING a trig equation?

• It means finding the values of x that will make the equation true. (Just as we did with algebraic equations!)

• Until now, we have worked with identities, equations that are true for ALL values of x. Now we’ll be solving equations that are true only for specific values of x.

Solving Trigonometric EquationsQuadrant IQuadrant II

Quadrant III Quadrant IV

Cosine

AllSine

Tangent

1800 - q1800 + q 3600 - q

q

12

12

32

13

32

3

12 1

12

Exact Values of Special Angles

Solving Trigonometric Equations

sinq 3

2

q 3

q 3

,23

1) tanq 1

q 4

q 4

,54

2)

Solve for q if 0 ≤ q < 2.

General Solutions22 , 2

3 3..., 2, 1,0,1,2

n n

n

q + +

- -

ReferenceAngle

ReferenceAngle

General Solutions

q 4+ 2n, 5

4+ 2n

n I

sin2 q 12

q 4

q 4

,34

,54

,74

3)

sinq 12

sinq 2

2

Solving Trigonometric Equations

Solve for q if 0 ≤ q < 2.

ReferenceAngle

sinq 12

4) 2tan 1 0x - 2tan 1x

2tan 1x tan 1x

3 5 7, , ,4 4 4 4

x

Q1 QII QIII QIV

5)

Solving Trigonometric Equations

Solve for q if 0 ≤ q < 2.

3 cscq - 2 03 cscq 2

cscq 23

q

3

, 23

6) 4 cosq+ 3 2 cosq + 22 cosq - 1

cosq -12

q

23

, 43

1. Try to get equations in terms of one trig function by using identities.

6. Try to get trig functions of the same angle. If one term is cos2q and another is cosq for example, use the double angle formula to express first term in terms of just q instead of 2q

3. Get one side equal to zero and factor out any common trig functions or reverse FOIL.

4. See if equation is in quadratic form and reverse FOIL. (Replace the trig function with x to see how it factors if that helps.)

5. If the angle you are solving for is a multiple of q, don't forget to add 2 to your answer for each multiple of q since q will still be less than 2 when solved for.

HELPFUL HINTS FOR SOLVING TRIGONOMETRIC EQUATIONS

2. Be on the look-out for ways to substitute using identities

7) 8)

NO solution forcos q = 3.

ReferenceAngle

Solving Trigonometric Equations

sin q 12 q

6

q 0 7

611

6, , ,

cosq 12

or

q

353

,

2 02sin sinq q+ sin ( sin )q q2 1 0+ sin sinq q + 0 2 1 0or

( cos )(cos )2 1 3 0q q- - 2 7 3 02cos cosq q- +

2 1 0 3 0cos cosq q- - orcosq 3

9) Solve 6 sin2 x + sin x – 1 = 0; 0º ≤ x < 360º

A quadratic equation!

It may help to abbreviate sin x with s:

i.e. 6s2 + s – 1 = 0

Factoring this: (3s – 1)(2s + 1)= 0

1 1sin or sin3 2

x x -∴

α = 19.47º α = 30º

So, x = (To nearest 0.1º.)19.5°, 160.5°, 210°, 330°.

You Try…Solve for y

03sin13sin12 2 +- yy

01sin33sin4 -- yyy

y

yy

-

43arcsin

43sin

3sin403sin4

y

y

yy

-

31arcsin

31sin

1sin301sin3

3398.0y 8481.0y

10)

01334

031312 2

--

+-

uu

uu

Solve for y: domain

We have to give all the answers

3398.0

8481.0

)2,0[

31

43 ?

?

294.28481.0 -

803.2339.0 -

11)The equation cannotbe factored. Therefore,use the quadratic equation to find the roots:

Reference Angles:

Solving Trigonometric Equations

tanq - -b b ac

a

2 42

tan ( ) ( ) ( )( )( )

q - - - - -1 1 4 3 1

2 3

2

3 1 02tan tanq q- -

tan . tan .q q- 0 43 0 76or

q 0 4061. q 0 6499.

q 2 7355 5 877 0 6499 3 7915. , . , . , .

3 tan2q - tanq - 1 0

Using a Graphing Calculator to Solve Trigonometric Equations

Therefore, q 0.654, 2.731, 3.796, and 5.873 .

2 2cos sin sin 0q q q- + Use the Pythagorean Identity to replace this with an equivalent expression using sine. 2 2cos 1 sinq q -

2 21 sin sin sin 0q q q- - + Combine like terms, multiply by -1 and put in descending order22sin sin 1 0q q- - Factor (think of sin q like x

and this is quadratic)

(2sin 1)(sin 1) 0q q+ - 1sin , sin 12

q q-

Set each factor = 0 and solve7 11, ,

6 6 2 q

12)

Solve for x where the domain is )2,0[ 2tansincos + xxx

35,

3;

21arccos

cos21;2

cos1

xx

xx

13)

2cos

sincos

2cossin

coscos

2cossinsincos

22

22

+

+

+

xxx

xx

xx

xxxx

Use a graphing utility to solve the equation. Express any solutions rounded to two decimal places.

3sin1722 - xxGraph this side as y1

in your calculatorGraph this side as y2

in your calculator

You want to know where they are equal. That would be where their graphs intersect. You can use the trace feature or the intersect feature to find this (or these) points (there could be more than one point of intersection).

There are some equations that can't be solved by hand and we must use a some kind of technology.

14)

3sin1722 - xx This was graphed on the computer with graphcalc, a free graphing utility you can download at www.graphcalc.com

After seeing the initial graph, lets change the window to get a better view of the intersection point and then we'll do a trace.

Rounded to 2 decimal places, the point of intersection is x = 0.53

check: 22 .53 17sin .53 3.066 3-

This is off a little due to the fact we approximated. If you carried it to more decimal places you'd have more accuracy.

YOU TRY…Solve

cos cos

cos cos: ( cos )(cos )

( cos ) (cos )

( cos ,cos ) (cos )

( , ) ( )

x x

x xfactor x x

x OR x

x x OR x

x n n OR x n

+ -

+ + + +

+ + -

- -

+ + +

2

22 3 12 3 1 0

2 1 1 02 1 0 1 0

12 1 122 42 2 23 3

xx 2sin23cos3 .1 +-0cos33)cos1(2 2 --- xx

2cot cos 2cotx x x2cot cos 2cot 0x x x-

2cot cos 2 0x x - 2cot 0 cos 2 0x or x -

3,2 2

x

2cos 2x 2cos 2x

cos 2x

x

Note: There is no solution here because 2 lies outside the range for cosine.

2)

Closure

Discuss the similarities and differences in the steps for solving a trigonometric equation versus solving a polynomial equation.

Solving Trigonometric Equations with Multiple Angles

1cos32

x

53 ,3 3

x

5,9 9

x

Solve:

Solution:

Since 3x refers to an angle, find the angles whose cosine value is ½.Now divide by 3 because it is angle equaling angle.

Notice the solutions do not exceed 2. Therefore,more solutions may exist.

Return to the step where you have 3x equalingthe two angles and find coterminal angles for those two.

7 11, ,3 3 53 ,

3 3x

Divide those two new angles by 3.7 11, ,9 9 5,

9 9x

1)

Solving Trigonometric Equations

The solutions still do not exceed 2. Return to 3x and find two more coterminal angles.

13 17, ,3 3 7 11, ,

3 3 53 ,

3 3x

Divide those two new angles by 3.13 17, ,9 9 7 11, ,

9 9 5,

9 9x

The solutions still do not exceed 2. Return to 3x and find two more coterminal angles.

19 23, ,3 3 13 17, ,

3 3 7 11, ,

3 3 53 ,

3 3x

Divide those two new angles by 3.19,913 17, ,

9 9 7 11, ,

9 9 5,

9 9x

Notice that 19/9 now exceeds 2 andis not part of the solution.

Therefore the solution to cos 3x = ½ is 5 7 11 13 17, , , , ,

9 9 9 9 9 9x

2) Solve the equations.

Example 3) Solve 2sin 3 42x+

0 .2

x2sin 1

21sin

2 2

x

x

5 and 6 6

5 or 2 6 2 6

5 or 3 3

x x

x x

x

y

Take the fourth root of both sides to obtain: cos(2x)= ±

23

From the unit circle, the solutions for 2q are 2q = ± + kπ, k any integer.

π 6

Example 4: Find all solutions of cos4(2x) = .9 16

Answer: q = ± + k ( ), for k any integer. 12

π 2

π

1 π 6-π 6

x = -23 x =

23

ππ

5) Solve 32

3 0tan x+

Example 6: Solve 3 + 5 tan 2x = 0; 0º ≤ x ≤ 360º.

Firstly we need to make tan 2x the subject of the equation:

The tangent of an angle is negative in the

- 5

3tan α 1 = 30.96°

The range must be adjusted for the angle 2x. i.e. 0° ≤ 2x ≤ 720°.

Hence: 2x =

x = 74.5°, 164.5°, 254.5°, 344.5°.

149.04°, 329.04°, 509.04°, 689.04°.

2nd and 4th quadrants.

tan 2x = – 35

Example 7: Solve 2 sin (4x + 90º) – 1 = 0; 0 < x < 90º 12

The sine of an angle is positive in the

The range must be adjusted for the angle 4x + 90º.i.e. 0º < 4x < 360º 90º < 4x + 90º < 450º

4x + 90º =

4x = 60º, 300º

x = 15º, 75º

-

21sin α 1 = 30º

150°, 390°

1st and 2nd quadrants.

sin (4x + 90º)

Solving Trigonometric Equations

24sin 2cos 1x x +

csc cot 1x x+

3sin 22

x -

Try these:

2cos2 2x

5.4218x

2x

2 5 5 11, , ,3 6 3 6

x

2x

1.

2.

3.

4.

Solution

Sect. 5-3#’s 61, 63, 65,

Evaluate.

a) tan2 116 b) sec2 2

3 -

33

2

13

- 2 2

4

Finding Exact Values

Solving Trigonometric Equationscos 1 sinx x+ 2 2cos 1 sinx x+

2 2cos 2cos 1 sinx x x+ +

Solve:

2 2cos 2cos 1 1 cosx x x+ + -22cos 2cos 0x x+

2cos cos 1 0x x + 2cos 0 cos 1 0x or x + cos 0x cos 1x -

3,2 2

x x X

Why is 3/2 removed as a solution?

26)

5.5 Trigonometric Equations• Objectives

–Find all solutions of a trig equation–Solve equations with multiple angles–Solve trig equations quadratic in form–Use factoring to separate different functions in

trig equations–Use identities to solve trig equations–Use a calculator to solve trig equations

Is this different that solving algebraic equations?• Not really, but sometimes we utilize trig

identities to facilitate solving the equation.• Steps are similar: Get function in terms of one

trig function, isolate that function, then determine what values of x would have that specific value of the trig function.

• You may also have to factor, simplify, etc, just as if it were an algebraic equation.

2cos 1 0q - 5)2cos 1x

1cos 2x

5,3 3

x

QI QIV

8)

ReferenceAngle

sin q 12

or sin q - 1

q

6

q

656

32

, ,

2 1 0 1 0sin sinq q- + or

2 1 02sin sinq q+ - ( sin )(sin )2 1 1 0q q- +

14)

Reference Angle:

Therefore:

sin q 13

4 1sin sinq q- 3 1sinq

q 0 3398.

q 0 3398 2 8018. .and