Trigonometric Equations

Edited by Mr. Francis Hung

Last Updated: 2013–03–12

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Trigonometric Equations

• sin x = sin x = or 180 –

• sin x = sin 30 x = 30 or 180 – 30 x = 30 or 150

• sin x = sin (–120) x = –120 or 180 –(–120) or –120+360 x = 300 or 240

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sin x = sin then x = or 180 –

360 45or 45180or 45 x

21

sin • x

315or 225 x

• sin x = –1

x = –90 or 180 – (–90) x = 270

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2or or 0 x

0 sin • xsin x = sin then x = or –

0.9 sin • x

12.12or 12.1or 1.12- x

5.16or 4.26 x

• sin x = 1.2

∵ –1 sin x 1

x has no solution

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• cos x = cos 130 x = 130 or 360 – 130 x = 130 or 230

cos x = cos then x = or 360 –

• cos x = –0.9

x = 154 or 360 – 154 x = 154 or 206

• cos x = –3

∵ –1 cos x 1

x has no solution5http:///www.hkedcity.net/ihouse/

fh7878/

cos x = cos then x = or 360 –

•cos x = cos (– 20) cos x = cos 20 x = 20 or 360 – 20 x = 20 or 340

• cos x = cos (–10) x = –10 or 360 – (–10) or 360 + (–10) or 10 x = 10 or 350

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32

or 3

4

x

34

cos cos•x

cos x = cos then x = or 2 –

3

42or

3

4

x

• cos x = tan 0.5c

cos x = 0.5463

x = 0.9929 or 2 – 0.9929

x = 0.993 or 5.297http:///www.hkedcity.net/ihouse/

fh7878/

31

tan • x

tan x = tan then x = or 180+

210or 30 x• tan x = –1

x = –45 or 180 + (–45) or 360 + (– 45) x = 135 or 315

• tan x = 5

x = 78.7 or 259

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tan x = tan then x = or 180+• sin x = –2cos x

tan x = –2

x = –63.4 or 180 + (–63.4) or 360 + (–63.4) x = 117 or 297

• tan x = –2 (sin 60 + 1)

tan x = –3.73

x = 105 or 285

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tan x = tan then x = or +

• tan x = –0.5

x = –0.464c or – 0.464c or 2 – 0.464c

x = 2.68 or 5.82

31

tan • x

62or

6or

6

x

611

or 6

5

x

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Exercise: solve the trigonometric equations1. sin x = sin(–15)

195 or 345

2. Answer in radians: sin x = 0.6

0.644 or 2.50

3. Answer in terms of :

4. sin x = 7no solution

5. cos x = cos(–330)30 or 330

6. cos x = 0x = 90 or 270

7. Answer in radians: cos x = –1/3

1.91 or 4.3723

sin x

35

or 3

4

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Exercise: solve the trigonometric equations

8. Answer in terms of :cos x = –1

9. Answer in terms of :cos x = –sin(3/4)3/4 or 5/4

10.Answer in terms of :

11.tan x = tan 5400, 180 or 360

12.3 sin x = 2 cos x33.7 or 214

13.Answer in terms of : tan x = –1x = 3/4 or 7/4

14.Answer in radians: tan x = 3

1.25 or 4.39

54

cos cosx

56

or 5

4

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More difficult examples1. cos 2x = cos 60

2x = 60, 300, 420, 660x = 30, 150, 210, 330

21

3

sin .2 x

135or 45 3x

(rejected) 540or 135 x

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More difficult examples3. cos 2x = cos (10 + x)

2x = 10 + x or 2x = 360 – (10 + x)

x = 10 or 116.67Is there any other solution between 0 and 360?236.67, 356.67

4. 2 cos2 – 3 cos + 1 = 0

(2 cos – 1)(cos – 1) = 0

cos = 0.5 or cos = 1

= 60, 300 or 0, 36014http:///www.hkedcity.net/ihouse/

fh7878/

More difficult examples5. 2 tan2 + tan – 1 = 0 (Answer in radians.)

(2 tan – 1)(tan + 1) = 0tan = 0.5 or tan = –1 = 0.464c, 3.61c or 3/4, 7/4

6. cos 3x = sin 2xcos 3x = cos(90 – 2x)3x = 90 – 2x or 3x = 360 – (90 – 2x)x = 18 or 270Is there any other solution between 0 and 360?90, 162, 234, 306

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More difficult examples7. 2 sin2 – cos – 1 = 0 (Answer in terms of .)

2(1 – cos2 ) – cos – 1 = 02 cos2 + cos – 1 = 0(2 cos – 1)(cos + 1) = 0cos = 0.5 or cos = –1 = /3, 5/3 or

8. sin tan + cos = 1 (Answer in terms of .)sin ( sin / cos ) + cos = 1sin2 + cos2 = cos cos = 1 = 0c or 2

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More difficult examples9. 3 – 2 sin cos – 4 sin2 = 0

3(sin2 + cos2 ) – 2 sin cos – 4 sin2 = 0

3 cos2 – 2 sin cos – sin2 = 0

3 – 2 tan – tan2 = 0

tan2 + 2 tan – 3 = 0

(tan + 3)(tan – 1) = 0

tan = –3 or tan = 1

= 108, 288 or 45, 225

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More difficult examples10. 12 sin – 5 cos = 13 (answer in radians.)

(12 sin – 5 cos )2 = 169 144 sin2 – 120sincos + 25cos2 =169(sin2+cos2)25 sin2 + 120 sin cos + 144 cos2 = 025 tan2 + 120 tan + 144 = 0(5 tan + 12)2 = 0tan = –12/5 = 1.97c, 5.11c

Check: when = 1.97c, LHS = 12 sin 1.97c – 5 cos 1.97c = 13 = RHSwhen = 5.11c, LHS = 12 sin 5.11c – 5 cos 5.11c = –13 RHS = 1.97c only

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SummaryIn degrees,

• sin x = sin then x = or 180 – • cos x = cos then x = or 360 – • tan x = tan then x = or 180 + In radians,

• sin x = sin then x = or – • cos x = cos then x = or 2 – • tan x = tan then x = or +

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