HONORS PRECALCULUS Prove the following identities- ( ) x x x x x x. cos x cos x cos x cos x 1 sin x cos x 1 sin x
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1 HONORS PRECALCULUS Prove the following identities-.) ( ) cos sin cos cos sin + sin sin + cos sin cos sin cos.).) ( ) ( sin) ( ) ( ) sin sin + + sin sin tan + sec + cos cos cos cos sin cos sin cos cos cos sin sin + cos cos + cos + cos + + csc cos + cos cos + cos + cos cos cos sin ( )( ) ( )( ) 4.) + + cos + + tan sin cos sin sin cos sin ( cos ) sin ( cos ) cos cos sec cos cos cos cos cos sin sin cos cot cos cos cos cos csc cos cot sin sin 5.) ( ) 6.) ( + ) ( + ) tan tan sec + tan sec tan sec sec + sec sec sec + sec tan tan
2 Trig Equations Worksheet 5.: Key. sin π 5π, 6 6. sin + sin 4π 5π, 5. cos cos π π, cot+ cot π 5π, 9. sec sec ±, π. tan π π, cos + cos 5π 7π, tan + tan π 7π, csc csc ± π π,. 4sec 8 sec ± π π 5π 7π,,, P age
3 Trig Identities: Worksheet.4. sec tan sin sec sin sin cos cos sin cos cos cos cos sec. + cos csc + cot sin cos + csc + cot sin sin. secθ sin θ tanθ + sin θ sin θ + sin θ 4. secθ tanθ sin cos cos sin θ 5. cos y sin y sin y ( ) sin y sin y sin y 6. csc θ tan θ tan θ sin θ sin θ tan θ P age
4 7. 8. tan θ sin θ tan θ sin θ csc θ tan θ( ) tan θ tan θcos θ tan θ sin θ tan θ tan θ sin θ sin θ csc θ sin θ 9. ( θ θ) ( θ θ) sin + cos + sin cos sin θ sin θ + ( θ ) sin + (). ( )( ) + tanθ + secθ + cscθ sin θ sin θ secθ cscθ secθ + cscθ. tanθ tanθ + + tan tan ( θ ) ( θ + ) tanθ tanθ + +. tan θ sin θ + tan θ ( ) sin sin θ ( θ ) P age
5 . cos + csc csc cot csc + cot sin cos ( csc cot )( csc + cot ) cos + sin sin ()( csc + cot ) csc + cot cos + csc cos cos + csc csc cos + cos cos ( )( ) 5. tanθ + + secθ cscθ sin y+ tan y sin y + secy sin y sin y + cosy + cos y sin y sin y cos y + cosy cos y + cos y sin y+ sin ycos y cos y + sin y( + cos y) sin y cos y + P age
6 5. and 5. Review: Trigonometric Equations and Trigonometric identities Prove the following trigonometric identities... + cos sin + csc sin + cos ( + cos ) + sin sin ( + cos ) + cos + cos + sin sin cos ( + ) + cos + sin cos ( + ) ( + ) ( + ) cos sin cos ( cos )( + cos ) csc sin + csc cos + cos + cos + cos cos csc sin. csc cos csc csc csc csc sin cos 4. ( )( ) ( csc )( sin ) ( csc )( cos ) cot + sin cot ( cos ) cot sin 5. sin csc + cot cos sin ( + cos ) ( cos )( + cos ) sin + sin cos cos sin + sin cos sin sin sin cos + sin sin cos csc + cot + sin sin 6. ( + sin)( + sin) ( sin )( + sin ) + sin + sin sin cos ( + sin) sin ( + sin) cos + sin + sin cos cos P age
7 Find all solutions to the following trigonometric equations 7. cos cos π 5π, 8. tan sin tan tan sin tan ( ) tan sin sin tan sin ±, π 9. sec 4 4 sec sec ± π 5π 7π π,,, cos cos ± π π 5π 7π,,, cos cos 4 cos ± π 5π 7π π,,, P age
8 HCP PRECALC NAME REVIEW NO CALCULATORS 5. Find the eact value of the following: tan ( 5 ) tan45 tan ( )( ) 4 tan ( 5 ) tan( 45 ) + tan45tan ( )( ) + + π 6. Use the appropriate Half-Angle Identity to find the eact value of sin 8 π + π π cos sin sin ± ± ± ± 8 4 π Note, we use the positive answer, because is in the first quadrant 8 Use identities to find all solutions in the interval,π ) sin cos sin, π, π, π,4 π,5π π π 4π 5π,,, π,, cos π π, π π π 4π π 5π,,,, π,,, cos+ cos cos( + ) + cos( ) cos cos sin sin + cos cos + sin sin cos cos cos cos π π 5π 7π π π,,,, π π 5π 7π,,, π π π 5π π 7π,,,,,
9 Use identities to find the general solution, i.e., ALL solutions, to the following. cos sin cos sin coscos sin sin sin cos coscos sin cos sin + sin cos cos + ( cos sin sin ) ( ) ( ) cos cos sin + sin ( ) cos cos + sin + sin ( ) ( ) cos + sin cos sin cos sin π π π 5π,, 6 6 π π 5π π 9.,,, 6 6 cos cos ± cos cos cos + cos cos cos + cos cos π π π, π π π,,. ( )( )
10 Prove the following identities:.. sin cos sin sin ( cos sin ) ( ) ( ) ( ) ( ) cos sin sin sin ( csc ) ( csc ) sin sin sin sin 4sin sin sin + sin sin sin + cos sin cos + cossin ( sin cos) cos+ cossin sin cos + cos sin sin csc sec csc csc sin csc sin sin cos sec
= tanθ 3) cos2 θ. = tan θ. = 3cosθ 6) sinθ + cosθcotθ = cscθ. = 3cosθ. = 3cosθ sinθ
PRE-CALCULUS/TRIGONOMETRY 3 Name 5.-5.5 REVIEW Date: Block Verify. ) cscθ secθ = cotθ 2) sec2 θ tanθ = tanθ 3) cos2 θ +sin θ = Use RIs sin θ = cotθ tan 2 θ tanθ = tan θ sin 2 θ +sin θ = Multiply by reciprocal
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