Barnett/Ziegler/Byleen Precalculus: Functions & Graphs, 4th Edition Chapter Five Trigonometric Functions Copyright © 1999 by the McGraw-Hill Companies, Inc.

Wrapping Function 5-1-48

Circular Functions If x is a real number and (a, b) are the coordinates of the circular point W(x), then: 5-2-49

Angles (a)  positive (b)  negative (c)  and  coterminal (a)  is a quadrantal (b)  is a third-quadrant (c)  is a second-quadrant angle angle angle 5-3-50-1

Angles (a) Straight angle (b) Right angle (c) Acute angle (d) Obtuse angle 5-3-50-2

Radian Measure 5-3-51

Trigonometric Functions with Angle Domains If q is an angle with radian measure x, then the value of each trigonometric function at q is given by its value at the real number x. Trigonometric Circular Function Function sin q = sin x cos q = cos x tan q = tan x csc q = csc x sec q = sec x cot q = cot x 5-4-52

Trigonometric Functions with Angle Domains Alternate Form If q 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 q, then: 5-4-53

Reference Triangle and Reference Angle 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. 5-4-54

30—60  and 45  Special Triangles ° (  /6) 45 ° 2 (  /4) 2 1 3 45 ° (  /4) 1 60 °  ( /3) 1 5-4-55

Trigonometric Functions with Angle Domains Alternate Form If q 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 q, then: 5-5-53

Graph of y = sin x Period: 2 Domain: All real numbers Range: [–1, 1] /2 Graph of y = sin x a b b P (cos x , sin x ) (0, 1) 1 x b Period: 2 Domain: All real numbers Range: [–1, 1] Symmetric with respect to the origin a  2  (–1, 0) a (1, 0) y = sin x = b (0, –1) 3  /2 y 1 x       –2 – 2 3 4 -1 5-6-56

Graph of y = cos x Period: 2 Domain: All real numbers Range: [–1, 1] /2 a b b Graph of y = cos x P (cos x , sin x ) (0, 1) 1 x b Period: 2 Domain: All real numbers Range: [–1, 1] Symmetric with respect to the y axis a  2  (–1, 0) a (1, 0) y = cos x = a (0, –1) 3  /2 y 1 x –2  –   2  3  4  -1 5-6-57

Graph of y = tan x 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 1 –2  –   2  x 5  3    3  5  – – – 2 2 2 2 2 2 –1 5-6-58

Graph of y = cot x 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 1 3    3  – – 2 2 2 2 x –2  –   2  –1 5-6-59

Graph of y = csc x y y = csc x sin 1 = y = sin x 1 x –2  –   2  –1  2  –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 5-6-60

Graph of y = sec x     y y = sec x cos 1 = y = cos x 1 x –2 – 2 –1  2  –1 Period: 2 Domain: All real numbers except /2 + k, k an integer Symmetric with respect to the y axis Discontinuous at x = /2 + k, k an integer Range: All real numbers y such that y  –1 or y  1 5-6-61

    Step 1. Find the amplitude | A |. Step 2. Solve Bx + C = 0 and = 2  : Bx + C = 0 and Bx + C = 2  C C 2 x = – x = – + B B B Phase shift Period C 2  Phase shift = – Period = B B The graph completes one full cycle as Bx + C varies from 0 to 2  — that is, as x varies over the interval é C C 2  ù ê – , – + ú B B B ë û é C C 2 ù Step 3. Graph one cycle over the interval ê – , – + ú . B B B ë û Step 4. Extend the graph in step 3 to the left or right as desired. 5-7-62

Graph of y = tan x 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 1 –2  –   2  x 5  3    3  5  – – – 2 2 2 2 2 2 –1 5-8-58

Graph of y = cot x 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 1 3    3  – – 2 2 2 2 x –2  –   2  –1 5-8-59

Graph of y = csc x y y = csc x sin 1 = y = sin x 1 x –2  –   2  –1  2  –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 5-8-60

Graph of y = sec x     y y = sec x cos 1 = y = cos x 1 x –2 – 2 –1  2  –1 Period: 2 Domain: All real numbers except /2 + k, k an integer Symmetric with respect to the y axis Discontinuous at x = /2 + k, k an integer Range: All real numbers y such that y  –1 or y  1 5-8-61

Facts about Inverse Functions 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–1 3. 4. 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)] = y for y in the domain of f–1 f–1[f(x)] = x for x in the domain of f 5-9-63

Inverse Sine Function Sine function     y y = sin x y = arcsin x æ –  2 1 x  2 –1 Sine function y –1 y = sin x y = arcsin x æ  ö y = sin x  1 ,  è 2 ø æ  2 ö –  2 1 , 1 è 2 ø (0,0) (0,0) x x –1 1  2 æ  ö – , –1 –1 æ  ö  è 2 ø –1 , – – è 2 ø 2 é   ù D OMAIN = ê – , ú D OMAIN = [–1, 1] ë 2 2 û é   ù R ANGE = [–1, 1] R ANGE = ê – , ú ë 2 2 û Restricted sine function Inverse sine function 5-9-64

Inverse Cosine Function y 1 x  –1 Cosine function y y = cos x = arccos –1 y y = cos x (–1,  )  (0,1) 1 è æ ø ö  2 ,0 è æ ø ö ,  2  2 x   2 –1 (1,0) (  , –1) x –1 1 D OMAIN = [0,  ] D OMAIN = [–1, 1] R ANGE = [–1, 1] R ANGE = [0,  ] Restricted cosine function Inverse cosine function 5-9-65

Inverse Tangent Function y y = tan x Tangent function 1  2 3  2 x 3  2 – –  2 –1 y y y = tan –1 x  æ  ö y = tan x = arctan x 1 ,  è 4 ø 2 –  2 æ  ö 1 , 1 è 4 ø –1 x x  2 æ  1 ö – , –1 –1 è 4 ø  2 æ  ö –1 , – – è 4 ø æ   ö D OMAIN = (–  ,  ) D OMAIN = ç – , ÷ è 2 2 ø æ   ö R ANGE = ç – , ÷ R ANGE = (–  ,  ) è 2 2 ø Restricted tangent function Inverse tangent function 5-9-66