Copyright © 2000 by the McGraw-Hill Companies, Inc. Barnett/Ziegler/Byleen Precalculus: A Graphing Approach Chapter Five Trigonometric Functions

Copyright © 2000 by the McGraw-Hill Companies, Inc. Wrapping Function

Copyright © 2000 by the McGraw-Hill Companies, Inc. If x is a real number and (a, b) are the coordinates of the circular point W(x), then: Circular Functions

Copyright © 2000 by the McGraw-Hill Companies, Inc.  Initial side Terminal side  Initial side Terminal side  Initial side Terminal side x y III IIIIV  Initial side Terminal side x y III IIIIV  x y III IIIIV  Initial side Terminal side  Initial side Terminal side (a)  positive (b)  negative (c)  and  coterminal (a)  is a quadrantal (b)  is a third-quadrant(c)  is a second-quadrant angle angle angle Angles (a)

Copyright © 2000 by the McGraw-Hill Companies, Inc. 180° 90°   (a) Straight angle (b) Right angle (c) Acute angle (d) Obtuse angle Angles (b)

Copyright © 2000 by the McGraw-Hill Companies, Inc. r O  r Radian Measure s

Copyright © 2000 by the McGraw-Hill Companies, Inc. (a,b) W(x) (1, 0) a b x units arc length xrad  Trigonometric Circular Function sin  = x cos  = x tan  = tan x csc  = x sec  = x cot  = cot x If  is an angle with radian measure x, then the value of each trigonometric function at    is given by its value at the real number x. Trigonometric Functions with Angle Domains

Copyright © 2000 by the McGraw-Hill Companies, Inc.  a b a b r a b  r a b r  a b a b If  is an arbitrary angle in standard position in a rectangular coordinate system and P(a, b) is a point r units from the origin on the terminal side of , then: Trigonometric Functions with Angle Domains Alternate Form tan  =, a  0 P(a, b)

Copyright © 2000 by the McGraw-Hill Companies, Inc. 1. To form a reference triangle for , draw a perpendicular from a point P(a, b) on the terminal side of  to the horizontal axis. 2. The reference angle  is the acute angle (always taken positive) between the terminal side of  and the horizontal axis. a b   a b P(a,b) Reference Triangle and Reference Angle

Copyright © 2000 by the McGraw-Hill Companies, Inc ° 60° (  /6) (  /3) ° ° (  /4) (  30  — 60  and 45  Special Triangles

Copyright © 2000 by the McGraw-Hill Companies, Inc. Right Triangle Ratios Hyp Opp Adj  0° <  < 90°

Copyright © 2000 by the McGraw-Hill Companies, Inc. 2   /2  3  (1, 0)(–1, 0) (0, –1) (0, 1) 0 1 P(cos x, sin x) a b b a x a b x y 1 0  2  3  4  –  –2  Period: 2  Domain: All real numbers Range: [–1, 1] Symmetric with respect to the origin Graph of y = sin x y = sin x = b

Copyright © 2000 by the McGraw-Hill Companies, Inc. 2   /2  3  (1, 0)(–1, 0) (0, –1) (0, 1) 0 1 P(cos x, sin x) a b b a x a b x y 1 0  2  3  4  –  –2  Period: 2  Domain: All real numbers Range: [–1, 1] Symmetric with respect to the y axis y = cos x = a Graph of y = cos x

Copyright © 2000 by the McGraw-Hill Companies, Inc. x y  2  –2  –  0 1 –1 5  2 3  2  2 – 5  2 – 3  2 –  2 Period:  Domain: All real numbers except  /2 + k , k an integer Range: All real numbers Symmetric with respect to the origin Increasing function between asymptotes Discontinuous at x =  /2 + k , k an integer Graph of y = tan x

Copyright © 2000 by the McGraw-Hill Companies, Inc.  2  –2  –  0 –  2 x y 1 –1 3  2  2 – 3  2 Period:  Domain: All real numbers except k , k an integer Range: All real numbers Symmetric with respect to the origin Decreasing function between asymptotes Discontinuous at x = k , k an integer Graph of y = cot x

Copyright © 2000 by the McGraw-Hill Companies, Inc. x y 1 –1  2  –  –2  0 y = sin x y = csc x sin x 1 = Period: 2  Domain: All real numbers except k , k an integer Range: All real numbers y such that y  –1 or y  1 Symmetric with respect to the origin Discontinuous at x = k , k an integer Graph of y = csc x

Copyright © 2000 by the McGraw-Hill Companies, Inc. x y 1 –1  2  –  –2  0 y = cos x y = sec x cos x 1 = Period: 2  Domain: All real numbers except  /2 + k , k an integerSymmetric with respect to the y axis Range: All real numbers y such that y  –1 or y  1Discontinuous at x =  /2 + k , k an integer Graph of y = sec x

Copyright © 2000 by the McGraw-Hill Companies, Inc. Step 1.Find the amplitude | A |. Step 2. Solve Bx + C = 0 and Bx + C = 2  : Bx + C =0 and Bx + C = 2  x = – C B x C B + 22 B Phase shift Period Phase shift = – C B Period = 2  B The graph completes one full cycle as Bx + C varies from 0 to 2  — that is, as x varies over the interval       – C B, – C B + 2  B Step 3.Graph one cycle over the interval       – C B, – C B + 22 B. Step 4.Extend the graph in Step 3 to the left or right asdesired

Copyright © 2000 by the McGraw-Hill Companies, Inc. DOMAINfRANGEf f DOMAINf –1 RANGEf –1 f x y f(x) f (y) For f a one-to-one function and f –1 its inverse: 1. If (a, b) is an element of f, then (b, a) is an element of f –1, and conversely. 2. Range of f = Domain of f –1 Domain of f = Range of f – If x = f –1 (y), then y = f(x) for y in the domain of f –1 and x in the domain of f, and conversely. 5. f [f –1 (y)] = yfor y in the domain of f –1 f –1 [f(x)] = xfor x in the domain of f x = f (y) –1 y= f(x) y x Facts about Inverse Functions

Copyright © 2000 by the McGraw-Hill Companies, Inc. x y 1 –1 –  2  2 Sine function     –  2, –1      2, 1 x y 1 –1  2 –  2 (0,0) y = sinx     –1, –  2     1,  2 x y 1–1  2 –  2 (0,0) y = sinx = arcsinx –1 D OMAIN =       –  2,  R ANGE = [–1, 1] Restricted sine function D OMAIN = [–1, 1] R ANGE =       –  2,  2 Inverse sine function 2 Inverse Sine Function

Copyright © 2000 by the McGraw-Hill Companies, Inc. x y 1 –1  y = cosx x y 1 –1  (0,1) ( , –1) 0  2      2,0 –1 y = cosx = arccosx –1  2 x y 1  (1,0) (–1,  ) 0     0,  2 Cosine function D OMAIN = [0,  ]D OMAIN = [–1, 1] R ANGE = [–1, 1]R ANGE = [0,  ] Restricted cosine functionInverse cosine function Inverse Cosine Function

Copyright © 2000 by the McGraw-Hill Companies, Inc. x y 1 –1  2 3  2 3  2 ––  2 y = tanx     –  4, –1      4, 1 x y 1 –1  2 –  2 y = tanx     –1, –  4     1,  4 y = tanx = arctanx –1 x y 1  2 –  2 Tangent function D OMAIN =       –  2,  2 R ANGE = (– ,  ) Restricted tangent function D OMAIN = (– ,  ) R ANGE =       –  2,  2 Inverse tangent function Inverse Tangent Function