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1 Chapter TRIGONOMETRIC FUNCTIONS Section. Angles. (a) 0 (b) 0. (a) 0 (b) 0. (a) (b). (a) (b). (a) (b). (a) (b) 9. (a) 9 (b) 9. (a) 0 (b) 0 9. (a) 0 (b) 0 0. (a) 0 0 (b) 0 0. (a) (b) (a) (b) ; ; ; 0.. ;.. 0 ; ; ; ; ;.. 0 ; (90x). (0 x). ( x0). ( x + 0) Copyright 0 Pearson Education, Inc.

2 Chapter Trigonometric Functions In exercises 9 9, answers may vary. 9. 0,0, 0, ,900, 0, ,0, 0, ,990, 90, n n n n n n n 0 = n n 0, or n 0 0. The answers to Exercises 0 and 0 give the same set of angles since 0 is coterminal with = Choices C and D are not coterminal with r. For Exercises 0, angles other than those given are possible Copyright 0 Pearson Education, Inc.

3 Section. Angles Copyright 0 Pearson Education, Inc.

4 Chapter Trigonometric Functions Copyright 0 Pearson Education, Inc.

5 Section. Angle relationships and Similar Triangles rotations in hr. or.0. sec. Section.. ; 0 ; 0 ;0 0 ;0 ; Angle Relationships and Similar Triangles. You can determine the measure of angle. Since the measure of angle is the same as measures of angles,, and, you know the measures of all eight angles.. ;. 9 ; 9. 0, 0, and 0. 0, 0, and 0. 0 ;0 ;0. 0 ;0 ;0 9.,, and ,, and. 9 ; 9. ;. ;. ; A triangle cannot have angles of measures and 00. The sum of the measures of these two angles is + 00 =, which exceeds 0.. A triangle cannot have two obtuse angles. Since an obtuse angle measures between 90 and 0, the sum of two obtuse angles would be between 0 and 0, which exceeds 0.. right; scalene. obtuse; scalene.. acute; equilateral.. acute; isosceles. 9. right; scalene. 0. obtuse; isosceles.. right; isosceles.. The triangle has a right angle with three unequal sides, so it is right and scalene.. The triangle has one obtuse angle and three unequal sides, so it is obtuse and scalene.. This triangle has three equal sides and all angles are acute, so it is acute and equilateral.. The triangle has three acute angles and two equal sides, so it is acute and isosceles.. This triangle has a right angle with three unequal sides, so it is right and scalene.. 9. Answers will vary. 0. Connect the right end of the semicircle to the point where the arc crosses the semicircle. Since the setting of the compass has never changed, the triangle is equilateral. Therefore, each of its angles measures 0.. A and P, B and Q, C and R. Corresponding sides are AC and PR, BC and QR, AB and PQ.. A and P, C and R, B and Q. AC and PR, CB and RQ, AB and PQ.. A and C, E and D, ABE and CBD. EB and DB, AB and CB, AE and CD. H and F, K and E, HGK and FGE. HK and FE, GK and GE, HG and FG.. Q is ; B = R = Copyright 0 Pearson Education, Inc.

6 Chapter Trigonometric Functions. P is ; A = N =. B is 0 ; A = M =. T is ; Z = W = 9. X = M = 0. T is 0 ; V is ; R = U = 9 In Exercises, corresponding sides of similar triangles are proportional. Other proportions are possible in solving these exercises.. a = 0; b =. a = 0; b = 0. a = ;. a =. x =. = m b =. 0 m. 0 ft m; 00 m 0. m.. ft. 0 ft. x = 0. y = Chapter Quiz (Sections..). (a) (b). ;. ;..,, and 0. 0 ; 0. (a).0 (b) 0 0. (a) 0 (b) 00 (c) 0 (d) º ft 0. (a) x= ; y= 0 (b) x = Section.. Trigonometric Functions. c».. m».. (a)»,000 mi (b) No. (a)»,,000 mi (b) Yes 9. (a)» 900 mi (b) No 0. (a)»,0,000 mi (b) Yes.. ; ; ; ; ;. (a) approximately (b) approximately 0 arc degrees. (a) CAG and HAD are equal and HD = (b) AGE and ADB are equal. (c) EF = BD = pace (d) CG = EG. (e) approximately the number of paces... ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; Copyright 0 Pearson Education, Inc.

7 Section. Trigonometri Functions.... ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; 9. ; 0; undefined; 0; undefined; 0. ; 0; undefined; 0; undefined;. 0; ;0; undefined; ; undefined. 0; ;0; undefined; ; undefined. ;0;undefined; 0;undefined;. ; 0; undefined; 0; undefined; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ;. Answers will vary. For any nonquadrantal angle θ, a point on the terminal side of θ will be of the form ( x, y ) where xy¹, 0. y r Now sinθ = and cscθ = both exist and r y are simply reciprocals of each other, and hence will have the same sign.. Answers will vary. r is the distance from (x, y) to the origin.. 0. tan θ and cot θ are positive; all other functions values are In Exercises, positive. r= x + y, which is. negative. negative. negative. negative 9. positive 0. negative. positive. negative. negative. negative. positive. positive. negative. negative 9. positive 0. positive. positive. positive. positive. positive ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ;. ; ; ; ; ; Copyright 0 Pearson Education, Inc.

8 Chapter Trigonometric Functions... ; ; ; ; ; ; ; ; ; ; ; ; ; ; ;. ; 0; undefined; 0; undefined;. 0; ;0; undefined; ; undefined undefined. undefined undefined. undefined Answers will vary undefined undefined undefined 0. undefined 0. They are equal. 0. They are equal. 0. The sines are negatives of each other. 0. The cosines are equal. In Exercises 0, make sure your calculator is in the modes indicated in the instructions. 0. about 0.90, and sin 0 is about T = T = 0. T =. decreases; increases.. decreases; decreases. Section. Using the Definitions of the Trigonometric Functions No. The value of cosθ cannot exceed.. Since tan 90 is undefined, it does not have a reciprocal.. cotθ =. tanθ 9. All are positive 0. All are positive. Tangent and cotangent are positive; all others are negative. Tangent and cotangent are positive; all others are negative. Sine and cosecant are positive; all others are. Sine and cosecant are positive; all others are. Cosine and secant are positive; all others are. Cosine and secant are positive; all others are Copyright 0 Pearson Education, Inc.

9 Section. Using the definitions of the Trigonometri Functions 9. Sine and cosecant are positive; all others are. cosine and secant are positive; all others are 9. All are positive. 0. All are positive.. I, II. I, IV. I. I. II. III. I. I 9. III 0. II. III, IV. II, IV. The answers to exercises and are the same because functions in exercise are the reciprocals of the functions in exercise.. cotθ > 0. Impossible. Impossible. Possible. Possible 9. Possible 0. Possible. Impossible. Impossible. Possible. Possible. Possible. Possible. possible. impossible 9. impossible 0. impossible For Exercises 9, remember that r is always positive ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; 9 ; 9 9 ; 0 ; ; ; ; ; ; ; ; ; 0 ; ; ; ; 9 ; ; 9 ; 9 ; ; ; ; ;. ; ; ; ; ; 9. 0.; 0.9; 0.;.009;.0; ; 0.; 0.;.9;.0; ( cotθ) = ( cscθ) or. cotθ + cot θ = csc θ.. The statement is false. For example, sin0 + cos0 = 0+ = ¹. ( ). The statement is false since sinθ for all θ.. negative. negative. positive. positive 9. negative 90. negative 9. negative 9. negative 9. positive 9. positive 9. negative 9. negative 9. positive 9. positive 99. negative 00. negative In quadrant II, the cosine is negative and the sine is positive. Copyright 0 Pearson Education, Inc.

10 0 Chapter Trigonometric Functions 0. III Chapter Review Exercises. complement of ; supplement of y = 0 ; x = 0. 0 in sec.. 90 in. sec ,. 0, 0, and 0. θ = β α. BA = 0. km. V = ; Z = ; m Y = 0 Since angle U corresponds to angle Y, the measure of angle U is 0.. N = ; R = ; M =. m = n = 0 9. p = q = proportional; equal. ft.. ; ;. 0; ; 0;.. ; ; ; ; ; ; ; ; ; ; undefined; ; 0 ; ; ; 0 undefined 9 9 ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ;. ; ; ; ; ;. The tangent and secant are undefined... ; ; ; ; ; sin θ = ;cos θ ; tanθ = cot θ = ;secθ = ; cscθ =. 0; ;0;undefined; undefined.. ;0;undefined;0;undefined; 9. (a) Impossible (b) Possible (c) Impossible 0... ; 9 ; ; ; ; ; 9 ; ; ; ; 9 ; 9 ; ; ;. ; ; ;. ; ; ; ;. ; ; ; ; ;. quadrant IV;. 0 yards. ; ; ; 9 9 ; Copyright 0 Pearson Education, Inc.

11 Chapter Test. 0 sec/yr 9. about 900 feet deep 0.,00 feet tall. Chapter Test. (a) (b). and. 0 and 0. The angles each measure 0. The angles each measure 0. The three angles measure 0, 0, and 0. The three angles measure 00, 0, and (a) 0 (b) 0 (c) in one second.. 0 ft, or 0 ft, in.. x= ; y=.. sinθ =; cosθ = 0; cotθ = 0; tanθ = undefined; 0 secθ = undefined; cscθ = 0 sin θ = ; cos θ = ; tanθ = ; cot θ = ; secθ = ; cscθ =. Column : ;0;undefined;0;undefined; Column : 0;;0;undefined;;undefined Column : ;0;undefined;0;undefined;. undefined. 9. (a) I (b) III and IV (c) III 0. (a) Impossible (b) Possible (c) Possible. secθ =. sin θ = ; cos θ = ; tanθ = ; cot θ = ; sec θ = ; cscθ =. 0 cos θ = ; 0 cot θ = ; cscθ = 0 tan θ = ; 0 0 sec θ = ; 0 Copyright 0 Pearson Education, Inc.