Compound Angles Higher Maths
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
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
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}
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
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
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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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) = sin180cos30 + 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
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
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
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
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
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
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 24. Simplify cos225 – sin225 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). 12 13 4