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Compound Angles

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  1. Compound Angles Higher Maths

  2. Compound Angles Click on an icon Trig Equations 1 Ans Ans Ans Exact Values Trig Equations 2 Sin (A+B), Sin (A-B) Higher trig. questions Trig Equations 3 Cos (A+B) , Cos (A-B) Using the four formulae

  3. 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. 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

  6. 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°}

  7. Trigonometric equations 3 Solve for 0 360o x 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

  8. 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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  12. 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) . = - 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 . = Continued on next slide

  13. 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

  14. 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. , find the exact value of cos (A-B). 3. If A and B are acute and 4. If A is acute and , find the exact value of cos2A. Continued on next slide

  15. 5sin2x = 7cosx • 5cos2x – 7cosx + 6 = 0 • 4cos2x – 10sinx -7 = 0 • 4sin2x = 3sinx • 8cos2x – 2cosx + 3 = 0 • 3cos2x + 7sinx – 5 = 0 • 6sin2x = 11sinx 5. Solve the equations for 6. Solve for a) b) 2sin2x +sinx = 0 c) cos2x – 4cosx = 5 Continued on next slide

  16. 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

  17. 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 20. Use the formula for sin (x+y) to show that x+y = 45. 1 x y 3 2 Continued on next slide

  18. 21. Use the formula for cos (x+y) to show that cos (x+y) = and A is obtuse and B is acute, find the exact values of a) sin2A b) cos(A-B) 22. If sin A = , sin B= , 23. Solve the equation sinxcos33 + cosxsin33 = 0.9 24. Simplify cos225 – sin225 12 25 Solve the equations a) 4sin2x = 5sinx b) cos2x + 6cosx + 5 = 0 26. The diagram shows two right angled triangles. Find the exact value of sin (x+y). 13 4 y x 3 x y 3 3 2