Compound Angles Higher Maths Compound Angles Click on

Compound Angles Higher Maths

Compound Angles Click on an icon Ans Trig Equations 1 Trig Equations 3 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

Trigonometric Equations 1 Solve the following equations for 0 < x < 360, x R 1. 2 sin 2 x + 3 cos x = 0 2. 3 cos 2 x - cos x + 1 = 0 3. 3 cos 2 x + cos x + 2 = 0 4. 2 sin 2 x = 3 sin x 5. 3 cos 2 x = 2 + sin x 6. 10 cos 2 x + sin x - 7 = 0 7. 2 cos 2 x + cos x - 1 = 0 8. 6 cos 2 x - 5 cos x + 4 = 0 9. 4 cos 2 x - 2 sin x - 1 = 0 10. 5 cos 2 x + 7 sin x + 7 = 0 Solve the following equations for 0 < q < 2 , q R 11. sin 2 q - sin q = 0 12. sin 2 q + cos q = 0 13. cos 2 q + cos q = 0 14. cos 2 q + 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 2 Use the formula Sin 2 x = 2 sinxcosx , Cos 2 x = 2 cos 2 x -1 = 1 – 2 sin 2 x to solve the following equations, for 0 < x < 360, x R 1 5 sin 2 x = 7 cosx 2 3 cos 2 x – 10 cosx + 7 = 0 3 2 cos 2 x + 4 sinx + 1 = 0 4 sin 2 x = sinx 5 cos 2 x + cosx = 0 6 5 cos 2 x + 11 sinx – 8 = 0 7 3 sin 2 x = 5 cosx 8 2 cos 2 x – 9 cosx – 7 = 0 9 2 cos 2 x – sinx + 1 = 0 10 2 sin 2 x = 3 sinx 11 3 cos 2 x – 7 cosx + 4 = 0 12 4 cos 2 x + 13 sinx – 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 360 o 1. 5 cos 2 x + sinx – 2 = 0 8. 2 cos 2 x + cosx – 3 = 0 2. 3 cos 2 x – 2 cosx + 3 = 0 9. 3 sin 2 x = sinx 3. 5 sin 2 x = 7 cosx 10. 7 cos 2 x -17 cosx + 1 = 0 4. cos 2 x + 4 sinx -1 = 0 11. cos 2 x – 8 cosx + 1 = 0 5. 7 sin 2 x = 13 sinx 12. 4 sin 2 x = 5 cosx 6. cos 2 x + sinx – 1 = 0 13, 8 cos 2 x + 38 cosx + 29 = 0 7. 3 cos 2 x + sinx – 1 = 0 14. 3 cos 2 x – 11 sinx – 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}

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 sin 210 Solution 1 sin 210 = sin(180 + 30 ) = sin 180 cos 30 + cos 180 sin 30 = 0. + (-1). = Worked example 2 By writing 315 as 360 - 45 , find the exact value of cos 315 Solution 2 cos 315 = cos(360 - 45 ) = cos 360 cos 45 + sin 360 sin 45 = 1. + 0. = 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 2. If A is obtuse and 3. If A and B are acute and 4. If A is acute and Continued on next slide , find the exact values of sin 2 A and cos 2 A. , find the exact value of cos (A-B). , find the exact value of cos 2 A.

5. Solve the equations for a)5 sin 2 x = 7 cosx b)5 cos 2 x – 7 cosx + 6 = 0 c)4 cos 2 x – 10 sinx -7 = 0 d)4 sin 2 x = 3 sinx e)8 cos 2 x – 2 cosx + 3 = 0 f)3 cos 2 x + 7 sinx – 5 = 0 g)6 sin 2 x = 11 sinx 6. Solve for a) b) 2 sin 2 x +sinx = 0 c) cos 2 x – 4 cosx = 5 Continued on next slide

7. Find the exact value of sin 45 + sin 135 + sin 225 8. Show that 9. Show that sin(x+30) – cos(x+60) = 3 sinx 10. Show that sin(x+60) – sin(x+120) = sinx 11. Prove that 12. Prove that (sinx + cosx)2 = 1 + sin 2 x 13. Prove that sin 3 xcosx + cos 3 xsinx = sin 2 x 14. By writing 3 x as 2 x + x show that sin 3 x = 3 sinx – 4 sin 3 x cos 3 x = 4 cos 3 x – 3 cosx Continued on next slide

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) cos 330 b) sin 210 c) sin 135 19. If sinx= and x is acute, find the exact values of a) sin 2 x b) cos 2 x c) sin 4 x 20. Use the formula for sin (x+y) to show that x+y = 45. 1 x 3 Continued on next slide y 2

x y 21. Use the formula for cos (x+y) to show that cos (x+y) = 3 3 22. If sin A = , sin B= 2 , and A is obtuse and B is acute, find the exact values of a) sin 2 A b) cos(A-B) 23. Solve the equation sinxcos 33 + cosxsin 33 = 0. 9 24. Simplify cos 225 – sin 225 25 Solve the equations a) 4 sin 2 x = 5 sinx b) cos 2 x + 6 cosx + 5 = 0 26. The diagram shows two right angled triangles. Find the exact value of sin (x+y). 12 13 4 x 3 y