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Honors Geometry Advanced Trigonometry Review Worksheet for Lessons 9-12 answers Evaluate. 1. sec 45 = 1 cos45 2 = 1 2 2 = 2 = 2 2. csc 120 = 1 sin120 3 = 1 2 2 = 3 2 3 = 3 3. cot 60 = 1 tan60 = 1 3 1 = 3 3 = 3 4. sec 90 = 1 cos90 = 1 (0) undefined 9. cot-1(3) = tan-1(1/3) = 18.4 10. sec-1(-1) = cos-1(1/-1) = cos-1(-1) = 180 11. csc-1( 2 ) = sin-1 = sin-1 1 2 2 2 = 45 12. cot-1(0) = tan-1(1/0) = 90 (because tangent is undefined at 90) Solve. 0 < < 360 13. sec = 2 cos = ½ ref : 60 I: = 60 IV : = 360 – 60 = 300 Solution : { 60, 300 } 14. csc = -½ sin = 1/-½ = -2 undefined 5 csc(-40) = 1 sin(-40) = -1.5557 solution : 6. cot 70 = 1 tan70 = 0.3640 15. cot = 1 tan = 1/1 = 1 7. sec-1(4) = cos-1(1/4) = 75.5 ref : 45 I: = 45 III : = 180 + 45 = 225 8. csc-1(-2) = sin-1(1/-2) = -30 Solution : { 45, 225 } Honors Geometry Advanced Trigonometry Review Worksheet for Lessons 9-12 answers 16. sec2 - 3sec + 2 = 0 (sec -1)(sec - 2) = 0 sec - 1 = 0 sec = 1 cos = 1 = 0 sec - 2 = 0 sec = 2 cos = ½ ref : 60 I: = 60 IV: = 360-60 = 300 = sin 1 1 cos = sin cos = tan 20. tan2 + sin2 + cos2 Solution : { 0, 60, 300 } = tan2 + (sin2 + cos2) 17. 3cot2 - 1 = 0 3cot2 = 1 cot2 = 1/3 tan2 = 3 tan = 3 = tan2 + 1 tan = 3 tan = - 3 ref : 60 ref : 60 I: = 60 II: = 180-60 III: = 180+60 = 120 = 240 IV: = 360-60 = 300 Solution : { 60, 120, 240, 300 } 18. csc2 = csc + 2 csc2 - csc - 2 = 0 (csc + 1)(csc - 2) = 0 csc + 1 = 0 or csc - 2 = 0 csc = -1 csc = 2 sin = -1 sin = ½ = 270 = 30, 150 Solution : { 30, 150, 270 } = sec2 21. cos(-) sin(-) = cos -sin = - cos sin = -cot 22. sec ∙sin(90-) 1 cos = cos = 1 23. (cos - 1)(cos + 1) (1 + sin)(1 - sin) = cos2 - 1 1 – sin2 Simplify. = -(1 – cos2) cos2 19. sin∙sec = -sin2 cos2 Honors Geometry Advanced Trigonometry Review Worksheet for Lessons 9-12 answers = -tan2 III: + 10 = 180 + 68.2 + 10 = 248.2 = 238.2 24. (sin + cos)2 – 1 cos∙cos(90-) = (sin2 + 2sin∙cos + cos2) - 1 cos∙sin = 2sin∙cos + (sin2+cos2) – 1 sin∙cos = 2sin∙cos + 1 sin∙cos = 2sin∙cos sin∙cos = 2 Solve. 0 < < 360 25. sin(-) = 1 -sin = 1 sin = -1 = 270 solution : { 270 } 26. sec(-) = - 2 sec() = - 2 1 2 cos = 2 2 ref : 45 II : = 135 III: = 225 solution : { 135, 225 } 27. tan( + 10) = 2.5 ref : 68.2 I: + 10 = 68.2 = 58.2 -1 solution : { 58.2, 238.2 } 28. csc( + 20) = -2 sin( + 20) = - ½ ref : 30 III: + 20 = 180 + 30 = 210 = 190 IV: + 20 = 360 – 30 = 330 = 310 solution : { 190, 310 } 29. cos(2) = ½ Since 0 < < 360, then 0 < 2 < 720 ref : 60 I: 2 = 60 or 2 = 60 + 360 = 420 IV: 2 = 360 – 60 = 300 or 2 = 300 + 360 = 660 So 2 = 60, 300, 420, or 660 If 2 = 60, then = 30 If 2 = 300, then = 150 If 2 = 420, then = 210 If 2 = 660, then = 330 solution : { 30, 150, 210, 330 } 30. cot( ½) = -1 Since 0 < < 360, then 0 < ½ < 180 tan( ½) = 1/-1 = -1 ref : 45 II: ½ = 180 – 45 = 135 IV: ½ = 360 – 45 = 315 But 0 < ½ < 180 !!! So, ½ = 135 Honors Geometry Advanced Trigonometry Review Worksheet for Lessons 9-12 answers If ½ = 135, then = 270 solution : { 270 } 31. sin(-3) = 0 -sin(3) = 0 sin(3) = 0 Since 0 < < 360, then 0 < 3 < 1080 Quadrant angle. = 0 or = 0 + 360 = 360 or = 360 + 360 = 720 = 180 or = 180 + 360 = 540 or = 540 + 360 = 900 So, 3 = 0, 180, 360, 540, 720, or 900 If 3 = 0, then = 0 If 3 = 180, then = 60 If 3 = 360, then = 120 If 3 = 540, then = 180 If 3 = 720, then = 240 If 3 = 900, then = 300 solution : {0,60,120,180,240,300} 32. tan(-5) = 1 -tan(5) = 1 tan(5) = -1 Since 0 < < 360, then 0 < 5 < 1800 Round 3 5 = 495 + 360 = 855 5 = 675 + 360 = 1035 Round 4 5 = 855 + 360 = 1215 5 = 1035 + 360 = 1395 Round 5 5 = 1215 + 360 = 1575 5 = 1395 + 360 = 1755 If 5 = 135, then = If 5 = 315, then = If 5 = 495, then = If 5 = 675, then = If 5 = 855, then = If 5 = 1035, then = If 5 = 1215, then = If 5 = 1395, then = If 5 = 1575, then = If 5 = 1755, then = 27 63 99 135 171 207 243 279 315 351 solution : {27, 63, 99, 135, 171, 207, 243, 279, 315, 351} 33. cos(100 - ) = 0.5 cos[-( -100)] = 0.5 cos( -100) = 0.5 ref : 60 I: -100 = 60 = 160 Since 0 < 5 < 1800, we can go around the circle five times IV: -100 = 300 = 400 Round 1 5 = 135 5 = 315 Since 400 > 360, = 400 – 360 = 40 Round 2 5 = 135 + 360 = 495 5 = 315 + 360 = 675 solution : {40, 160}