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Barnett/Ziegler/Byleen Precalculus: Functions & Graphs, 4th Edition Chapter Five Trigonometric Functions Copyright © 1999 by the McGraw-Hill Companies, Inc. Wrapping Function v v 2 2 1 v 2 1 1 3 (1, 0) 0 (1, 0) 0 u (1, 0) 0 u –1 –1 –2 –2 v u –3 –2 –1 v |x| A(1, 0) P 0 u A(1, 0) u 0 |x| P (a) x > 0 (b) x < 0 5-1-48 Circular Functions If x is a real number and (a, b) are the coordinates of the circular point W(x), then: v sin x = b 1 csc x = b cos x = a 1 sec x = a b tan x = a a a 0 cot x = b b0 a0 (a, b) W(x) (1, 0) u b0 5-2-49 Angles Terminal side Terminal side Initial side Initial side (a) positive (b) negative Terminal side IV (a) is a quadrantal angle I II Initial side Initial side x III y II I Terminal side III Initial side (c) and coterminal y y II Terminal side x Terminal side I x Initial side IV (b) is a third-quadrant angle III IV (c) is a second-quadrant angle 5-3-50-1 Angles 180 ° (a) Straight angle 1 ( 2 rotation) 90 ° (b) Right angle 1 ( 4 rotation) (c) Acute angle (0° < < 90°) (d) Obtuse angle (90° < < 180°) 5-3-50-2 Radian Measure s s = r radians Also, s = r r r O s =r r r = r = 1 radian O r 1 radian 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 Function Circular Function b (a, b) sin = sin x cos = cos x tan = tan x csc = csc x sec = sec x cot = cot x W(x) x rad x units arc length a (1, 0) 5-4-52 Trigonometric Functions with Angle Domains Alternate Form b 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: b a P ( a, b a ) b r r = a2 + b2 > 0; P(a, b) is an arbitrary point on the terminal side of , (a, b) (0, 0) b r b a b sin = r a cos = r b tan = a , a 0 a P(a, b ) r csc = b , b 0 r sec = a , a 0 a cot = b , b 0 a r a b P ( a, b ) 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. b 2. The reference angle is the acute angle (always taken positive) between the terminal side of and the horizontal axis. a a b (a, b) (0, 0) is always positive P(a, b) 5-4-54 30—60 and 45 Special Triangles 30 ° ( /6) 45 ° ( /4) 2 2 3 1 45 ° ( /4) 1 60 ° ( /3) 1 5-4-55 Trigonometric Functions with Angle Domains Alternate Form b 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: b a P ( a, b a ) b r b r = a2 + b2 > 0; P(a, b) is an arbitrary point on the terminal side of , (a, b) (0, 0) b r a b sin = r a cos = r b tan = a , a 0 a P(a, b ) r csc = b , b 0 r sec = a , a 0 a cot = b , b 0 a r a b P ( a, b ) 5-5-53 /2 a b b P(cos x , sin x ) (0, 1) 1 Graph of y = sin x x Period: 2 b a (–1, 0) 0 a (1, 0) 2 Domain: All real numbers Range: [–1, 1] y = sin x = b (0, –1) 3 /2 Symmetric with respect to the origin y 1 –2 – 0 2 3 4 x -1 5-6-56 /2 a b b P(cos x , sin x ) (0, 1) 1 Graph of y = cos x x b a (–1, 0) 0 a Period: 2 2 (1, 0) Domain: All real numbers Range: [–1, 1] Symmetric with respect to the y axis y = cos x = a (0, –1) 3 /2 y 1 –2 – 0 2 3 4 x -1 5-6-57 Graph of y = tan x y Period: Domain: All real numbers except /2 + k , k an integer 1 –2 – 5 2 – – 3 2 – 2 –1 2 Range: All real numbers 2 0 3 2 5 2 x Symmetric with respect to the origin Increasing function between asymptotes Discontinuous at x = /2 + k , k an integer 5-6-58 Graph of y = cot x y Period: Domain: All real numbers except k , k an integer 3 – 2 –2 – 2 1 – 2 0 –1 Range: All real numbers 3 2 2 x Symmetric with respect to the origin Decreasing function between asymptotes Discontinuous at x = k , k an integer 5-6-59 Graph of y = csc x y y = csc x = y = sin x 1 sin x 1 –2 – 0 2 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 5-6-60 Graph of y = sec x y y = sec x = 1 cos x 1 –2 – 0 –1 y = cos x 2 x Period: 2 , Domain: All real numbers except /2 + k Discontinuous at x = k an integer /2 + k, k an integer Symmetric with respect to the y axis Range: All real numbers y such that y –1 or y 1 5-6-61 Graphing y = A sin(Bx + C) and y = A cos(BX + C) Step 1. Find the amplitude | A |. Step 2. Solve Bx + C = 0 and Bx + C = 2 : Bx + C = 0 C x = –B and Bx + C = 2 C 2 x = –B + B Phase shift C Phase shift = – B Period 2 Period = B The graph completes one full cycle as Bx + C varies from 0 to 2 — that is, as x varies over the interval C C 2 – , – B+ B B Step 3. C C 2 Graph one cycle over the interval – B , – B + B . Step 4. Extend the graph in step 3 to the left or right as desired. 5-7-62 Graph of y = tan x y Period: Domain: All real numbers except /2 + k , k an integer 1 –2 – 5 2 – – 3 2 – 2 –1 2 Range: All real numbers 2 0 3 2 5 2 x Symmetric with respect to the origin Increasing function between asymptotes Discontinuous at x = /2 + k , k an integer 5-8-58 Graph of y = cot x y Period: Domain: All real numbers except k , k an integer 3 – 2 –2 – 2 1 – 2 0 –1 Range: All real numbers 3 2 2 x Symmetric with respect to the origin Decreasing function between asymptotes Discontinuous at x = k , k an integer 5-8-59 Graph of y = csc x y y = csc x = y = sin x 1 sin x 1 –2 – 0 2 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 5-8-60 Graph of y = sec x y y = sec x = 1 cos x 1 –2 – 0 –1 y = cos x 2 x Period: 2 , Domain: All real numbers except /2 + k Discontinuous at x = k an integer /2 + k, k an integer Symmetric with respect to the y axis 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. DOMAIN f RANGE f f x f ( x) f –1( y ) y f –1 RANGE f –1 5. f[f–1(y)] = y f–1[f(x)] = x DOMAIN f –1 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. y y = f (x) for y in the domain of f–1 for x in the domain of f x = f –1( y) x 5-9-63 Inverse Sine Function y – 1 2 –1 2 Sine function y y = sin x – – , –1 2 x , 1 2 1 2 (0,0) 2 –1 DOMAIN = – 2 , 2 RANGE = [–1, 1] Restricted sine function x y y = sin –1 x = arcsin x 1 , 2 2 (0,0) –1 –1 , – 2 x 1 – 2 DOMAIN = [–1, 1] RANGE = – 2 , 2 Inverse sine function 5-9-64 Inverse Cosine Function y 1 x –1 Cosine function y = arccos x y = cos x 1 0 –1 y y = cos –1 x (0,1) ,0 2 2 (–1, ) x 2 0 , 2 (1,0) ( , –1) –1 0 DOMAIN = [0, ] RANGE = [–1, 1] Restricted cosine function 1 x DOMAIN = [–1, 1] RANGE = [0, ] Inverse cosine function 5-9-65 Inverse Tangent Function y y = tan x 2 1 3 – 2 – 2 3 2 Tangent function x –1 y y = tan –1x = arctan x y = tan x – 2 , 1 4 x 2 1 – , –1 4 –1 y 2 1 , 4 –1 1 x –1 , – – 4 2 DOMAIN = – 2 , 2 RANGE = (– ,) Restricted tangent function DOMAIN = (– , ) RANGE = – 2 , 2 Inverse tangent function 5-9-66