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PRECALCULUS PREAP CALENDAR UNIT 7 January 5 – DECEMBER 18 Date Assignment Tuesday 1-5 Wednesday 1-6 Thursday 1-7 Friday 1-8 Monday 1-11 Tuesday 1-12 Graphing Inverse Trig Functions and Day 1 of Inverse Values Wednesday 1-13 Thursday 1-14 Friday 1-15 Reference Inverse Values Day 2 Analytical Trigonometry Simplifying Trig Expressions Simplifying Trig Expressions More Simplifying Trig Expressions Intro Proving Trig Identities Proving Trig Identities More Proving Trig Identities Review Test over Inverse Functions and Trig Identities 1 INVERSE SINE PARENT GRAPH Graph y = sin −1 ( x ) (the inverse function of sin x ) Alternate notation is y = arcsin ( x ) For y = sin −1 ( x ) , answer the following questions. 1) What is the domain? 2) What is the range? 3) So what angles are allowed with this function? Examples: Graph each function using your knowledge of transformations. State the domain and range. 1) y sin −1 ( x − 1) = 2) y = sin −1 ( 2 x ) Domain: Range: Domain: Range: y y 2π 2π 3π 2 3π 2 π π π 2 π 2 x −4 −3 −2 −1 1 2 3 4 x −4 −3 −2 −1 1 −π 2 −π 2 −π −π −3π 2 −3π 2 −2π −2π 2 3 4 2 = 3) y arcsin x − π = 4) y 4 Domain: Range: 1 −1 sin ( − x ) 2 Domain: Range: y y 2π 2π 3π 2 3π 2 π π π 2 π 2 x −4 −3 −2 1 −1 2 3 4 x −4 −3 −2 −1 1 −π 2 −π 2 −π −π −3π 2 −3π 2 −2π −2π 2 3 4 ------------------------------------------------------------------------------------------------------------------------------INVERSE COSINE PARENT GRAPH Graph y = cos −1 ( x ) (the inverse function of cos x ) Alternate notation is y = arccos ( x ) For y = cos −1 ( x ) , answer the following questions. 1) What is the domain? 2) What is the range? 3) So what angles are allowed with this function? 3 Examples: Graph each function using your knowledge of transformations. State the domain and range. = 1) y cos −1 x + π 2) y = − cos −1 ( x ) 2 Domain: Range: Domain: Range: y y 2π 2π 3π 2 3π 2 π π π 2 π 2 x −4 −3 −2 −1 1 2 4 3 −4 −3 −2 −1 1 −π 2 −π 2 −π −π −3π 2 −3π 2 −2π −2π y cos −1 ( − x ) + 4) = 3) y 2arccos ( x + 1) = Domain: x Range: 2 3 4 π 4 Domain: Range: y y 2π 2π 3π 2 3π 2 π π π 2 π 2 x −4 −3 −2 −1 1 2 3 4 x −4 −3 −2 −1 1 −π 2 −π 2 −π −π −3π 2 −3π 2 −2π −2π 2 3 4 4 INVERSE TANGENT PARENT GRAPH Graph y = tan −1 ( x ) (the inverse function of tan x ) Alternate notation is y = arctan ( x ) For y = tan −1 ( x ) , answer the following questions. 1) What is the domain? 2) What is the range? 3) So what angles are allowed with this function? Examples: Graph each function using your knowledge of transformations. State the domain and range. = 1) y tan −1 x − 3π 2 2) y = 2 tan −1 ( x ) Domain: Range: Domain: Range: _____ y y 2π 2π 3π 2 3π 2 π π π 2 π 2 x −4 −3 −2 −1 1 2 3 4 x −4 −3 −2 −1 1 −π 2 −π 2 −π −π −3π 2 −3π 2 −2π −2π 2 3 4 5 = 3) y 2arctan x − π 4) y = − tan −1 ( x + 1) 4 Domain: Range: Domain: Range: y y 2π 2π 3π 2 3π 2 π π π 2 π 2 x −4 −3 −2 −1 1 2 3 x 4 −4 −3 −2 −1 1 −π 2 −π 2 −π −π −3π 2 −3π 2 −2π −2π 2 3 4 GRAPHING INVERSE TRIG FUNCTIONS Find the domain, range, and sketch a complete graph of each function. Inverse functions are denoted by y = sin −1 x or by y = A rcsin x . 2) y = cos–1(x) - π 1) y = sin –1(3x) 5 y = 3 arccos(2x-4) 6) y = tan –1(x-1) + π 7) y = – arc sin x + 8) y = 3 cos–1 (x-2) 9) y = - 2 1 tan–1 (x-1) 4 3) y = arc sin (x+1) 4) π 2 x 3 y = 2 sin –1 ( ) 10) y = tan–1 x +1 6 NOTES REVIEW FINDING INVERSE TRIG VALUES AND ANGLES FIRST GROUP y = sin −1 x π π − 2, 2 y = csc −1 x π π − 2, 2 , y ≠ 0 y = tan −1 x π π − 2, 2 Sample Problems Triangle problems Draw a reference triangle to determine the value of each function. 3 Angle Problems 2) cot sin −1 − 1) sin arctan 5 4 sin −1 ( 1 10 3 ) 2 3) cos ( arctan x ) 5 arc tan(-1) 6 arcsin(tan 3π ) 4 SECOND GROUP y = cos −1 x [0, π ] y = cot −1 x (0, π ) y = sec −1 x [0, π ] y ≠ π 2 Sample Problems Triangle Problems Draw a reference triangle to determine the value of each function. 3 4) sin arccos 5 5) 5 sin cos −1 − 5 6) tan ( arcsin 2x ) 7 Angle Problems 4 cos −1 ( 3 ) 2 5 arc cot(-1) 6 arcsin(cos 3π ) 4 Summary of quadrant locations for inverse functions. Examples: Find the exact values without using a calculator. 3 cos −1 − = __________ 2 1. sin −1 (1) = ________ 3. 1 sin −1 ________ 2 5. tan −1 − 3 = ________ 6. 2 csc −1 = __________ 3 7. Arc csc 2 = ____________ 8. sec−1 9. sec −1 − 2 = _____________ 10. 1 A rc cot − = _______________ 3 ( ( ) ) 2. 4. 1 A rccos = __________ 2 ( 2 ) = ____________ 8 INVERSE VALUES AND ANGLES Find the exact value of the expression using radicals or radians if necessary. 5 1. sec sin −1 7 3 2. cot sec −1 − 2 3. csc ( tan −1 ( −5 ) ) π 4. sec −1 sin − 6 4π 5. sin −1 sin 3 Composition of Trig Values 5 6. sec csc −1 − 2 Find the exact values of each expression using radicals or radians if necessary. 4 4 5 1. tan cos −1 2. cos arctan 3. sin tan −1 − 5 3 12 π 6. arccos sin 6 7π 9. sin −1 cos 6 4 4. sec arcsin − 7 4π 7. cos −1 sin 3 5. sin −1 ( cos ( 0 ) ) 10. tan −1 ( cos (π ) ) 4π 11. tan −1 tan − 3 4 13. sin sec −1 − 3 π 14. arcsec sec − 3 16. tan arcsec − 2 17. sin ( sec −1 ( −4 ) ) 18. sec ( csc −1 ( −3) ) 20. arcsin ( cos (π ) ) 7 21. tan sin −1 − 5 ( 19. ( )) csc ( cot −1 ( 2 ) ) 5 8. cot csc −1 − 3 3 12. cos arcsin − 5 3π 15. sin −1 cot 4 9 INVERSE TRIG PROBLEMS Evaluate each of the following expressions. You are finding the value of each trig expression. 15 17 1) cos sin −1 6) cot tan −1 10 9) csc tan −1 −15 8 −1 3 13) csc tan −1 5 13 1 6 5) sin csc −1 5 2) sin cos −1 10) tan sec −1 −13 12 3) cos sin −1 −3 5 12 7) sec cot −1 5 11) tan sec −1 − 13 3 3 4) sin cos −1 3 5 8) tan csc −1 3 12) cos cot −1 −3 2 − 10 14) cot sin −1 10 Write and algebraic expression for the given expression. 15) tan ( arctan x ) 16) sin ( arccos x ) 17) cos ( arcsin 2x ) 18) sec ( arctan 3x ) x 19) tan arc sec 20) cot ( arc cot x ) 3 Part II Use a reference triangle to find the value. 1 1) sin arccos 2 1 4) cos arcsin 4 −3 2) sin arccos 5 3) tan arcsin 2 5) sec ( arc csc a ) x 6) cot arc csc 2 3 Find the angle. 7) arcsin( −1) −1 10) arccos 2 3π 13) arccos cot 2 − 2 8) arcsin 2 9) arccos( −1) π 11) arccos cos 3 3π 14) arcsin tan 4 4π 12) arcsin sin 3 15) arcsin sin −2π 3 10 Fundamental Trigonometric Identities Reciprocal Identities: Quotient Identities: Pythagorean Identities: Negative Angles Trig Identities Worksheet Day 1 Simplifying Expressions Simplify the given expression to a single trig term or a single number. Circle your answer. 1) cos2 θ + sin2 θ cos θ 3) 1 sec θ 2) sin θ ( cot ( −θ ) ) cos θ (1 + cot θ )(1 − cos θ ) 4) ( cos β )( sec 5) tan φ csc φ 6) sin θ ( csc θ − sin θ ) 7) cot θ sec θ 8) sec2 β (1 − sin2 β ) 9) cot φ sin φ 10) 1 + tan2 θ 1 + cot2 θ 11) sec θ − ( tan θ sin θ ) 12) cos2 θ sin θ 2 2 2 2 β − 1) + sin θ 11 sin θ + tan θ 1 + sec θ 13) (1 − sin θ )(1 + tan θ ) 16) ( sin θ + cos θ )2 + ( sin θ − cos θ )2 18) sin2 θ + cos θ cos θ 21) sec2 θ sec2 θ − 1 22) tan2 θ + 1 tan θ csc2 θ 24) 2 + cot2 θ −1 csc2 θ 25) cot2 θ − ( cos 4 θ csc2 θ ) 2 2 14) 19) 17) 15) sec θ tan θ − cos θ cot θ cos β csc β 1 1 + 2 sec θ csc2 θ 20) 23) sec θ sin θ tan θ + cot θ cot θ + tan θ csc θ + 1 PROVING TRIG IDENTITIES 1) sin x + cos x cot x = csc x 2) cos x csc x = cot x 3) 2 cos2 x - sin 2 x + 1 = 3 cos 2 x 4) sin x(sec x - csc x) = tan x - 1 5) 1 1 + = 2 sec2 x 1 + sin x 1 − sin x 6) sin2 x + cos x = 1 1 + cos x 7) sin x cot x + cos x = 2 cot x sin x 8) sin x cos x 1 = 2 1 − 2 sin x cot x − tan x 12 9) 10) 1 + tan2 x = csc2 x tan2 x 1 - tan x 1 + tan x = cot x - 1 cot x + 1 11) 1 − sin2 x = cot2 x 1 − cos2 x 12) sec x = sin x tan x + cot x 13) sin2 x = sec2 x − 1 cos2 x 14) sin x - sin x cos x 1 - cos x 1 + cos x = csc x + csc x cos2 x VERIFY THE FOLLOWING IDENTITIES. 1) csc2 x(1 − cos2 x) = 1 2) (sin x + cos x)2 − (sin x − cos x)2 = 4 sin x cos x 3) sin x(csc x + sin x sec2 x) = sec2 x 4) cot2 x += 5 csc2 x + 4 5) cos 4 x − cos2 x = sin 4 x − sin2 x 6) sinxtanx + cosx = secx 7) − cos2 x sin x − csc x = sin x 8) 1 1 + = cos x − sec x sec x cos x 9) sinx + cosxcotx = cscx 10) (sin x − cos x)2 = sec x − 2sin x cos x 11) sec x = sin x tan x + cot x 12) cos x 1 + sin x = 1 − sin x cos x 13 PRECALCULUS PREAP REVIEW Part 1 Find the domain and range for the following inverse functions. 1) = f ( x ) 2arcsin x − 2π 1 f ( x ) = arccos3 x 2 2) 3) = f ( x ) tan −1 ( 3 x ) + π 2 Part 2 Sketch the graph 4) y = 3sin −1 2 x 5) y = 2arctan x 1 = y arccos x + π 6) 2 Part 3 Simplify the following expressions. Give exact values. 2 2 7) arccos 11) 3 tan cos −1 2 8) 12) tan −1 ( −1) 3 sin cot −1 4 9) π cos −1 cos − 4 10) π sin −1 sin 5 13) tan ( sec −1 3x ) 14) x csc arccos 3 Part 4 Simplify the expression sec x tan x + cot x 15) 1 1 + 2 sec x csc 2 x 16) 19) 1 + tan x 1 + cot x 20) sin x sec x 17) (1 − tan 2 x )(1 − sin 2 x ) 21) sec x − sin 2 x sec x 18) sec x − sin x tan x 22) cos 2 x 1 − sin x Part 5 Prove the identities 23) sin x ( sec x − csc x ) =tan x − 1 24) 1 − cos x sin x + = 2csc x sin x 1 − cos x 25) tan x sin x tan x − sin x = tan x + sin x tan x sin x 14