= tanθ 3) cos2 θ. = tan θ. = 3cosθ 6) sinθ + cosθcotθ = cscθ. = 3cosθ. = 3cosθ sinθ

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1 PRE-CALCULUS/TRIGONOMETRY 3 Name 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 Cancel Factor DOS sin θ = cotθ tan θ = tan θ Cancel sin θ Use QI ( sin θ)(+sin θ) +sin θ = = cotθ tan θ = tan θ sin θ = sin θ cot θ = cotθ 4) sec2 θ+csc 2 θ = csc 2 θsec 2 θ 5) 2cos2 θ sin 2 θ+ = 3 6) + cotθ Divide through Change to sin/cos (Use QI) sec 2 θ csc 2 θsec 2 θ + csc2 θ csc 2 θsec 2 θ = 2cos 2 θ ( cos 2 θ)+ = 3 + Cancel Distribute negative Multiply + = 2cos 2 θ +cos 2 θ+ = 3 csc 2 θ sec 2 + cos2 θ θ Use RIs numerator Get common denominators sin 2 θ + cos 2 θ = 3cos 2 θ = 3 + cos2 θ Cancel Combine over common denom. = 3 = 3 sin 2 θ+cos 2 θ Use RI cscθ

2 7) +secθ tanθ+ 8) sin4 θ cos 4 θ = 2sin 2 θ 9) tan 2 θ( + cot 2 θ) = Change to sin/cos Factor DOS Distribute + + sin 2 θ (sin 2 θ cos 2 θ)(sin 2 θ + cos 2 θ) = tan 2 θ + tan 2 θcot 2 θ = Get common denom. Use RI + + (sin 2 θ cos 2 θ)() = 2sin 2 θ tan 2 θ + = Combine over common denom. + + sin 2 θ cos 2 θ = 2sin 2 θ sec 2 θ = Cancel denoms Which one doesn t belong? () Use RI + + sin2 θ ( sin 2 θ) = 2sin 2 θ GCF in denom. Distribute negative + (+) sin2 θ + sin 2 θ = 2sin 2 θ Cancel Use RI cscθ 2 sin2 θ = 2sin 2 θ sin 2 θ = cos 2 θ sin 2 θ = sin 2 θ sin 2 θ sin 2 θ sin 2 θ 0) tanθ cotθ = sec2 θ csc 2 θ ) ( ) = Divide through tanθ cotθ = sec2 θ csc 2 θ cos 2 θ 2 + sin 2 θ + 2 = Change to sin/cos (Use QIs) FOIL = sec 2 θ csc 2 θ cos 2 θ + sin 2 θ = Multiply by the reciprocal Cancel and simplify = cos 2 θ sin 2 θ sec2 θ csc 2 θ Use RIs sec 2 θ csc 2 θ = sec 2 θ csc 2 θ = sec 2 θ csc 2 θ =

3 2) ( cos 2 θ)( + cos 2 θ) = 2sin 2 θ sin 4 θ 3) tanθ cotθ tanθ+cotθ = 2sin2 θ (sin 2 θ)( + cos 2 θ) = 2sin 2 θ sin 4 θ Which one doesn t belong () (sin 2 θ)( + ( sin 2 θ)) = 2sin 2 θ sin 4 θ Combine like terms inside ( )s Change to sin/cos + = 2sin 2 θ Get common denominators + = 2sin 2 θ (sin 2 θ)(2 sin 2 θ) = 2sin 2 θ sin 4 θ Distribute sin 2 2sin 2 θ sin 4 θ = 2sin 2 θ sin 4 θ **Keep in mind you could have distributed and then used which one doesn t belong sin 2 θ+cos 2 θ = 2sin 2 θ Cancel common denominator sin 2 θ+cos 2 θ = 2sin2 θ = 2sin 2 θ Which one doesn t belong () sin 2 θ ( sin 2 θ) = 2sin 2 θ Distribute the negative and combine like terms 2sin 2 θ = 2sin 2 θ Use sum AND difference formulas to find the exact values of each function. 4) cos SUM cos(240+45) = cos 240 cos45 sin240 sin45 = ref = 60 QIII (T) (, = DIFFERENCE cos(33045) = cos 330 cos45 +sin330 sin45 = = QIV (C) ref = 30 ( 3, ) 2 2

4 5) sin SUM sin(300+45) = sin 300 cos45 + cos300 sin45 = = QIV (C) ref = 60 (, 3 ) 2 2 DIFFERENCE sin(39045) = sin 390 cos45 cos390 sin45 = Co-term = 30 QI (A) ref = 30 ( 3, ) 2 2 = 2 6 Use sum or difference formulas to find the exact values of each function. 6) cos 265 cos55 + sin265 sin55 7) sin305cos25 + cos305sin25 Compare formulas cos difference cos x cos y + sin x sin y cos 265 cos55 + sin265 sin55 Compare formulas sin sum sin x cos y + cos x sin y sin305cos25 + cos305sin25 Go backwards (x = 265, y = 55) Go backwards (x = 305, y = 25) cos(x y) = cos(265 55) = cos(20) sin( ) =sin(330) QIII (T) ref : 30 QIV (C) ref : 30 cos(20) = 3 2 Verify the following using the sum or difference formulas. sin(330)= 2 8) cos (90 x) = sin x 9) sin (270 + x) = -cos x Use cos difference cos 90 cos x + sin 90 sin x = sin x Use sin sum sin 270 cos x + cos 270sin x = - cos x (0) cos x + () sin x = sin x (-) cos x + (0)sin x = - cos x sin x = sin x - cos x = - cos x

5 If cos x = 5 3 and x is in the IV Quadrant, find double angle value: x cos x =, so x = 3 and r = 5 r QIV (+, ) QIV (C) so sin x and tan x are negative! 20) sin 2x 2) cos 2x 22) tan 2x Solve the following trigonometric equations. Calculate all answers over [0,360 ] and [0, 2]. 23) 2 cos x + = 0 24) 4 sin 2 x 2 = 0 25) 2 cos 2 x cos x = 0 x = 3π 4, 5π 4 x = π 3, π 2, 3π 2, 5π 3 x = π 4, 3π 4, 5π 4, 7π 4 26) 2 sin 2 x 4 sin x + 2 = 0 27) tan 2 x - 3 tan x = 0 28) 2 cos 2 x + 3 sin x 3 = 0 x = π 2 x = 0. π 3, π, 4π 3, 2π x = π 6, π 2, 5π 6