Transcript Compound angles

Compound Angles
Higher Maths
Compound Angles
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Ans
Trig Equations 1
Trig Equations 3
Trig Equations 2
Ans
Ans
Sin (A+B), Sin (A-B)
Exact Values
Cos (A+B) , Cos (A-B)
Higher trig. questions
Using the four formulae
Trigonometric Equations 1
Solve the following equations for 0 < x < 360, x  R
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
2sin 2x + 3cos x = 0
3cos 2x - cos x + 1 = 0
3cos 2x + cos x + 2 = 0
2sin 2x = 3sin x
3cos 2x = 2 + sin x
10cos2 x + sin x - 7 = 0
2cos 2x + cos x - 1 = 0
6cos 2x - 5cos x + 4 = 0
4cos 2x - 2sin x - 1 = 0
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.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
{90, 229, 270, 311}
{48, 120 , 240, 312}
{71,120 , 240 , 289}
{0, 41 , 180 , 319}
{19 , 161 , 210 , 330 }
{37, 143, 210, 330}
{41, 180, 319}
{48, 104, 256, 312}
{30, 150, 229, 311}
{233, 307}
{0, /3 ,  , 5/3 , 2}
{ /2 , 7/6 , 3 /2 , 11 /6}
{ /3, , 5 /3}
{ /2 , 7/6 , 11/6}
Trig. Equations 2
Use the formula Sin2x = 2sinxcosx , Cos2x = 2cos2x -1 = 1 – 2sin2x
to solve the following equations, for 0 < x < 360, x  R
1
5sin2x = 7cosx
2
3cos2x – 10cosx + 7 = 0
3
2cos2x + 4sinx + 1 = 0
4
sin2x = sinx
5
cos2x + cosx = 0
6
5cos2x + 11sinx – 8 = 0
7
3sin2x = 5cosx
8
2cos2x – 9cosx – 7 = 0
9
2cos2x – sinx + 1 = 0
10
2sin2x = 3sinx
11
3cos2x – 7cosx + 4 = 0
12
4cos2x + 13sinx – 9 = 0
Trig Equations (2) - Solutions
Question
Solution
1
{ 44°, 90°, 136°, 270°}
2
{ 48°, 312°}
3
{ 210°, 330°}
4
{ 60°, 180°, 300°}
5
{ 60°, 180°, 300°}
6
{ 30°, 37°, 143°, 150°}
7
{ 56°, 90°, 124°, 270°}
8
{ 221°, 139°}
9
{ 49°, 131°, 270°}
10
{ 41°, 180°, 319°}
11
{ 80°, 280°}
12
{ 39°, 90°, 141°}
Trigonometric equations 3
Solve for 0  x
 360o
1. 5cos2x + sinx – 2 = 0
8. 2cos2x + cosx – 3 = 0
2. 3cos2x – 2cosx + 3 = 0
9. 3sin2x = sinx
3. 5sin2x = 7cosx
10. 7cos2x -17cosx + 1 = 0
4. cos2x + 4sinx -1 = 0
11. cos2x – 8cosx + 1 = 0
5. 7sin2x = 13sinx
12. 4sin2x = 5cosx
6. cos2x + sinx – 1 = 0
13, 8cos2x + 38cosx + 29 = 0
7. 3cos2x + sinx – 1 = 0
14. 3cos2x – 11sinx – 8 = 0
Trig Equations 3 - Solutions
1. SS = {37,143,210,330}
3. SS = {44,90,136,270}
5. SS = {0,22,180,338,360}
7. SS = {42,138,210,330}
9.
SS = {0,80,180,280,360}
11. SS = {90,270}
13. SS = {151,209}
2.
SS = {71,90,270,289}
4.
SS = {0,180,360}
6.
SS = {0,30,150,180,360}
8. SS = {0,360}
10. SS = {107,253}
12. SS = {39,90,141,270}
14. SS = {236,270,304}
Continued on next slide
Continued on next slide
Continued on next slide
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
=
0.
= - 1
2
3
2
+ (-1) .
1
2
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. 1 + 0. 1
2
2
1
=
2
Continued on next slide
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 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 sin A 
2. If A is obtuse and sin A 
4
, find the exact values of sin2A and cos2A
5
5
, find the exact values of sin2A and cos2A.
13
3. If A and B are acute and sin A 
4. If A is acute and cos A 
Continued on next slide
1
1
, cos B 
, find the exact value
2
3
of cos (A-B).
8
, find the exact value of cos2A.
17
5. Solve the equations for
0  x  360
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
6. Solve for
0  x  2
a) 2 cos2 x  cos x  1  0
b) 2sin2x +sinx = 0
c) cos2x – 4cosx = 5
Continued on next slide
7. Find the exact value of sin45 + sin135 + sin225
8. Show that

1
sin(x  )  (cos x  3 sin x)
6
2
9. Show that sin(x+30) – cos(x+60) = 3sinx
10. Show that sin(x+60) – sin(x+120) = sinx
11. Prove that tan x  tan y 
sin(x  y )
cos x cos y
12. Prove that (sinx + cosx)2 = 1 + sin2x
13. Prove that sin3xcosx + cos3xsinx = 1 sin2x
2
14. By writing 3x as 2x + x show that
sin3x = 3sinx – 4sin3x
cos3x = 4cos3x – 3cosx
Continued on next slide
2 tan x
15. Using the fact that tan x  sin x , show that
 sin 2 x
2
cos x
1  tan x
16.
Prove that (cosx + cosy)2 + (sinx + siny)2 = 2[1+cos(x+y)]
17. Work out the exact values of
19. If sinx=
a) cos330 b) sin210 c) sin135
5
and x is acute, find the exact values of
13
a) sin2x
b) cos2x
c) sin4x
20. Use the formula for sin (x+y) to show that x+y = 45.
1
x
y
3
Continued on next slide
2
21. Use the formula for cos (x+y) to show that
cos (x+y) = 1
26
22. If sin A =
x y
3
3
2
1
1
, sin B= , and A is obtuse and B is acute,
2
3
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
0  x  360
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
x
3
y