1. Compound Angles
Higher Maths

2. Compound Angles Click on an icon Trig Equations 1 Trig Equations 2 Ans
Sin (A+B), Sin (A-B)
Exact Values
Cos (A+B) , Cos (A-B)
Higher trig. questions
Using the four formulae

3. Solve the following equations for 0 < q < 2 , q  R
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

4. 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}

5. 4cos2x + 13sinx – 9 = 0 12 3cos2x – 7cosx + 4 = 0 11 2sin2x = 3sinx 10
Trig. Equations 2
Use the formula Sin2x = 2sinxcosx , Cos2x = 2cos2x -1 = 1 – 2sin2x
to solve the following equations, for 0 < x < 360, x  R
4cos2x + 13sinx – 9 = 0 12
3cos2x – 7cosx + 4 = 0 11
2sin2x = 3sinx 10
2cos2x – sinx + 1 = 0 9
2cos2x – 9cosx – 7 = 0 8
3sin2x = 5cosx 7
5cos2x + 11sinx – 8 = 0 6
cos2x + cosx = 0 5
sin2x = sinx 4
2cos2x + 4sinx + 1 = 0 3
3cos2x – 10cosx + 7 = 0 2
5sin2x = 7cosx 1

6. 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 4
{ 210°, 330°} 3
{ 48°, 312°} 2
{ 44°, 90°, 136°, 270°} 1
Solution
Question

7. 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
cos2x -17cosx + 1 = 0
11. cos2x – 8cosx + 1 = 0
12. 4sin2x = 5cosx
13, 8cos2x + 38cosx + 29 = 0
14. 3cos2x – 11sinx – 8 = 0

8. Trig Equations 3 - Solutions
1. SS = {37,143,210,330} SS = {71,90,270,289}
3. SS = {44,90,136,270} SS = {0,180,360}
5. SS = {0,22,180,338,360} SS = {0,30,150,180,360}
7. SS = {42,138,210,330} SS = {0,360}
SS = {0,80,180,280,360} SS = {107,253}
11. SS = {90,270} SS = {39,90,141,270}
13. SS = {151,209} SS = {236,270,304}

9. Continued on next slide

11. Continued on next slide

13. Continued on next slide

15. By writing 210 as 180 + 30 , find the exact value of sin210
Exact Values
Worked example 1
By writing 210 as 180 + 30 , find the exact value of sin210
Solution 1
sin210 = sin(180 + 30)
= sin180cos30 + cos180 sin30 =
+ (-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 = =
Continued on next slide

16. Use the previous ideas to find the exact values of the following
1. sin 150 cos 225 sin 240
4. cos 300 sin 120 cos 135
7. sin 135 cos 210 sin 315

17. done the course work on trigonometry.
Higher Trigonometry Questions
This set of questions would be suitable as revision for pupils who have
done the course work on trigonometry.
1. If A is acute and
, find the exact values of sin2A and cos2A
2. If A is obtuse and
, find the exact values of sin2A and cos2A.
3. If A and B are acute and
, find the exact value
of cos (A-B).
4. If A is acute and
, find the exact value of cos2A.
Continued on next slide

18. 5. Solve the equations for 5sin2x = 7cosx 5cos2x – 7cosx + 6 = 0
4cos2x – 10sinx -7 = 0
4sin2x = 3sinx
8cos2x – 2cosx + 3 = 0
3cos2x + 7sinx – 5 = 0
6sin2x = 11sinx a)
b) 2sin2x +sinx = 0
c) cos2x – 4cosx = 5
6. Solve for
Continued on next slide

19. 7. Find the exact value of sin45 + sin135 + sin225
8. Show that
Show that sin(x+30) – cos(x+60) = 3sinx
10. Show that sin(x+60) – sin(x+120) = sinx
11. Prove that
12. Prove that (sinx + cosx)2 = 1 + sin2x
13. Prove that sin3xcosx + cos3xsinx =
sin2x
14. By writing 3x as 2x + x show that
sin3x = 3sinx – 4sin3x
cos3x = 4cos3x – 3cosx
Continued on next slide

20. Prove that (cosx + cosy)2 + (sinx + siny)2 = 2[1+cos(x+y)]
15. Using the fact that
, show that
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=
and x is acute, find the exact values of
a) sin2x b) cos2x c) sin4x 1 2 3 y x
20. Use the formula for sin (x+y) to show that x+y = 45.
Continued on next slide

21. Find the exact value of sin (x+y).
21. Use the formula for cos (x+y) to show that
cos (x+y) = 2 3 y x
22. 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
Simplify cos225 – sin225
25 Solve the equations
a) 4sin2x = 5sinx
b) cos2x + 6cosx + 5 = 0
The diagram shows two right angled triangles.
Find the exact value of sin (x+y). 12 13 4