Compound Angles

Higher Maths

Compound Angles

Trig Equations 1 Ans Ans

Ans

Trig Equations 2

Trig Equations 3

Sin (A+B), Sin (A-B)

Cos (A+B) , Cos (A-B)

Using the four formulae

Exact Values

Higher trig. questions

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Trigonometric Equations 1

Solve the following equations for 0 < x < 360, x R

1.      2sin 2x + 3cos x = 0 2.      3cos 2x - cos x + 1 = 0 3.      3cos 2x + cos x + 2 = 0 4.      2sin 2x = 3sin x 5.      3cos 2x = 2 + sin x 6.      10cos2 x + sin x - 7 = 0 7.      2cos 2x + cos x - 1 = 0 8.      6cos 2x - 5cos x + 4 = 0 9.      4cos 2x - 2sin x - 1 = 0 10.    5cos 2x + 7sin x + 7 = 0

Solve the following equations for 0 < q < 2 , q R 11.      sin 2q - sin q = 0 12.      sin 2q + cos q = 0 13.      cos 2q + cos q = 0 14.      cos 2q + sin q = 0

Trig Equations 1 - Solutions.

1.      {90, 229, 270, 311} 2.      {48, 120 , 240, 312}3.      {71,120 , 240 , 289}4.      {0, 41 , 180 , 319}5.      {19 , 161 , 210 , 330 }6.      {37, 143, 210, 330} 7.      {41, 180, 319} 8.      {48, 104, 256, 312} 9.      {30, 150, 229, 311} 10.      {233, 307} 11.      {0, /3 , , 5/3 , 2} 12.      { /2 , 7/6 , 3 /2 , 11 /6}13.      { /3, , 5 /3} 14.      { /2 , 7/6 , 11/6}

Trig. Equations 2Use the formula Sin2x = 2sinxcosx , Cos2x = 2cos2x -1 = 1 – 2sin2xto solve the following equations, for 0 < x < 360, x R

4cos2x + 13sinx – 9 = 012

3cos2x – 7cosx + 4 = 011

2sin2x = 3sinx10

2cos2x – sinx + 1 = 09

2cos2x – 9cosx – 7 = 08

3sin2x = 5cosx7

5cos2x + 11sinx – 8 = 06

cos2x + cosx = 05

sin2x = sinx4

2cos2x + 4sinx + 1 = 03

3cos2x – 10cosx + 7 = 02

5sin2x = 7cosx1

Trig Equations (2) - Solutions

{ 39°, 90°, 141°}12

{ 80°, 280°}11

{ 41°, 180°, 319°}10

{ 49°, 131°, 270°}9

{ 221°, 139°}8

{ 56°, 90°, 124°, 270°}7

{ 30°, 37°, 143°, 150°}6

{ 60°, 180°, 300°}5

{ 60°, 180°, 300°}4

{ 210°, 330°}3

{ 48°, 312°}2

{ 44°, 90°, 136°, 270°}1

SolutionQuestion

Trigonometric equations 3

Solve for 0 x 360o

1. 5cos2x + sinx – 2 = 0

2. 3cos2x – 2cosx + 3 = 0

3. 5sin2x = 7cosx

4. cos2x + 4sinx -1 = 0

5. 7sin2x = 13sinx

6. cos2x + sinx – 1 = 0

7. 3cos2x + sinx – 1 = 0

8. 2cos2x + cosx – 3 = 0

9. 3sin2x = sinx

10. 7cos2x -17cosx + 1 = 0

11. cos2x – 8cosx + 1 = 0

12. 4sin2x = 5cosx

13, 8cos2x + 38cosx + 29 = 0

14. 3cos2x – 11sinx – 8 = 0

Trig Equations 3 - Solutions

1. SS = {37,143,210,330} 2. SS = {71,90,270,289} 3. SS = {44,90,136,270} 4. SS = {0,180,360} 5. SS = {0,22,180,338,360} 6. SS = {0,30,150,180,360} 7. SS = {42,138,210,330} 8. SS = {0,360} 9. SS = {0,80,180,280,360} 10. SS = {107,253} 11. SS = {90,270} 12. SS = {39,90,141,270} 13. SS = {151,209} 14. SS = {236,270,304}

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2

32

1

2

1

Exact Values Worked example 1 By writing 210 as 180 + 30 , find the exact value of sin210

Solution 1 sin210 = sin(180 + 30)

= sin180cos30 + cos180 sin30

= 0 . + (-1) .

= -

2

1

2

1

2

1

Worked example 2 By writing 315 as 360 - 45 , find the exact value of cos315

Solution 2 cos315 = cos(360 - 45)

= cos360 cos45 + sin360 sin45 = 1 . + 0 .

=

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Use the previous ideas to find the exact values of the following 1. sin 150 2. cos 225 3. sin 240 4. cos 300 5. sin 120 6. cos 135 7. sin 135 8. cos 210 9. sin 315

Higher Trigonometry QuestionsThis set of questions would be suitable as revision for pupils who have

done the course work on trigonometry.

1. If A is acute and 5

4sin A , find the exact values of sin2A and cos2A

2. If A is obtuse and 13

5sin A , find the exact values of sin2A and cos2A.

3. If A and B are acute and 3

1cos,

2

1sin BA , find the exact value

of cos (A-B).

4. If A is acute and 17

8cos A , find the exact value of cos2A.

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3600 x5. Solve the equations for a)5sin2x = 7cosx

a)5cos2x – 7cosx + 6 = 0

a)4cos2x – 10sinx -7 = 0

a)4sin2x = 3sinx

a)8cos2x – 2cosx + 3 = 0

a)3cos2x + 7sinx – 5 = 0

a)6sin2x = 11sinx

20 x 01coscos2 2 xx a)

b) 2sin2x +sinx = 0

c) cos2x – 4cosx = 5

6. Solve for

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7. Find the exact value of sin45 + sin135 + sin225

8. Show that )sin3(cos2

1)

6sin( xxx

9. Show that sin(x+30) – cos(x+60) = 3sinx

10. Show that sin(x+60) – sin(x+120) = sinx

11. Prove that

yx

yxyx

coscos

)sin(tantan

12. Prove that (sinx + cosx)2 = 1 + sin2x

13. Prove that sin3xcosx + cos3xsinx = 2

1 sin2x

14. By writing 3x as 2x + x show thatsin3x = 3sinx – 4sin3x

cos3x = 4cos3x – 3cosx

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x

xx

cos

sintan x

x

x2sin

tan1

tan22

15. Using the fact that , show that

16. Prove that (cosx + cosy)2 + (sinx + siny)2 = 2[1+cos(x+y)]

17. Work out the exact values of a) cos330 b) sin210 c) sin135

19. If sinx=13

5 and x is acute, find the exact values of

a) sin2x b) cos2x c) sin4x

1

23yx

20. Use the formula for sin (x+y) to show that x+y = 45.

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21. Use the formula for cos (x+y) to show that cos (x+y) =

26

1

23

yx

3

2

1

3

122. If sin A = , sin B= , and A is obtuse and B is acute,

find the exact values of a) sin2A b) cos(A-B)

23. Solve the equation sinxcos33 + cosxsin33 = 0.9

24. Simplify cos225 – sin225

25 Solve the equations 3600 x a) 4sin2x = 5sinx

b) cos2x + 6cosx + 5 = 0

26. The diagram shows two right angled triangles. Find the exact value of sin (x+y).

12

13

4

3x

y