0 Capter Trigonometry 70. f 8 7 8 Vertical asymptote: 8 0 y 7 0 7 8 9 9 ± 8 y Slant asymptote: ± 89 ;.,. y 7 8 y-intercept: 0, 8 -intercept:.8, 0 Section. Rigt Triangle Trigonometry You sould know te rigt triangle definition of trigonometric functions. sin cos yp yp (d) csc yp (e) sec yp (f) tan cot You sould know te following identities. sin csc csc sin (d) sec (e) tan cos cot (f) (g) () cot cos tan sin (i) cos sin cos sec cot tan sin cos (j) tan sec (k) cot csc and 90, You sould know tat two acute angles are complementary if and tat cofunctions of complementary angles are equal. You sould know te trigonometric function values of 0,, and 0, or be able to construct triangles from wic you can determine tem. Vocabulary Ceck ypotenuse acent ypotenuse. (i) sec (v) (ii) (iv) (iii) csc (vi) osite cot acent osite acent osite osite (iv) (iii) (v) (i) (vi) (ii) acent tan ypotenuse sin ypotenuse cos. osite; acent; ypotenuse. elevation; depression
Section. Rigt Triangle Trigonometry. yp 8 00 0 8 sin yp 0 cos yp 8 0 tan 8 csc yp 0 sec yp 0 8 cot 8. b 9 sin yp csc cos yp sec tan cot yp yp. 9 8 8 00 0 9 sin yp 9 cos 0 yp tan 9 0 csc yp 9 sec yp 0 cot 0 9. yp sin csc yp cos sec yp tan cot yp yp. 8 sin yp cos yp tan csc yp sec yp cot sin yp cos yp tan csc yp sec yp cot Te function values are te same since te triangles are similar and te corresponding sides are proportional.
Capter Trigonometry. 8 7. yp 8 89 7 sin yp 8 7 cos yp 7 tan 8 csc yp 7 8 sec yp 7 cot 8 yp 7. 7 sin yp 7 8 7 cos 7. yp 7 7 tan 7. 8 csc yp 7 7 8 sec yp 7 7 7. cot 7. 8 Te function values are te same because te triangles are similar, and corresponding sides are proportional. 7. sin yp csc yp cos yp sec yp tan cot. 0.7. sin 0.7 yp. cos yp. tan 0.7 csc yp. 0.7 sec yp. cot 0.7 Te function values are te same since te triangles are similar and te corresponding sides are proportional. 8. yp sin yp cos yp tan csc yp sec yp cot yp sin cos csc sec tan cot Te function values are te same because te triangles are similar, and corresponding sides are proportional.
Section. Rigt Triangle Trigonometry 9. Given: sin yp 0. Given: cos 7 yp 7 cos 7 yp tan 7 7 csc yp sec yp 7 7 cot 7 7 7 sin yp 7 tan csc yp 7 7 sec yp 7 cot 7. Given: sec yp. Given: cot yp sin yp cos yp tan csc yp cot sin yp cos yp tan csc yp sec yp. Given: tan. Given: sec yp yp yp 0 sin 0 yp 0 cos 0 yp 0 csc yp 0 sec yp 0 cot 0 sin yp cos yp tan csc yp cot
Capter Trigonometry. Given: cot yp. Given: csc 7 yp 7 7 7 7 yp sin yp cos yp tan csc yp sec yp sin yp 7 cos 7 yp 7 tan 7 7 7 sec yp 7 77 7 7 cot 7 7. 0 0 0 0 80 radian sin 0 yp 8. 80 radian cos yp 9. 80 tan 0 0. 80 sec yp. 0 cot 0 radian. csc yp 80 radian 0. 80 cos 0 yp. sin 80 yp. cot radian 80. 0 0 tan 0 0 radian 80
Section. Rigt Triangle Trigonometry 7. sin 0, cos 0 8. sin 0, tan 0 tan 0 sin 0 cos 0 csc 0 sin 0 sin 0 cos 0 cot 0 tan90 0 tan 0 cos 0 sin 0 (d) cot 0 cos 0 sin 0 cos 0 sin 0 tan 0 (d) cot 0 tan 0 9. csc, sec sin csc cos sec tan sin cos (d) sec90 csc 0. sec, tan cos sec cot tan cot90º tan (d) sin tan cos. cos. tan sec cos sin cos sin sin 8 9 cot tan cos sec tan tan90º cot tan sin cot cos sin (d) csc cot (d) sin90 cos. tan cot tan tan. cos sec cos cos
Capter Trigonometry. tan cos sin cos cos sin. cot sin cos sin cos sin 7. cos cos cos sin cos cos sin 8. sin sin sin cos 9. sec tan sec tan sec tan 0. sin cos sin sin tan tan sin sin sin. sin cos sin cos cos sin sin cos sin cos sin cos csc sec. tan cot tan tan cot tan tan cot cot cot csc. sin 0 0.7. tan. 0.8 cos 80 0.7 Note: cos 80 sin90 80 sin 0 cot. 0.8 tan.. sin. 0.8 csc.. sin.. cos 8 cos 0 8 0.998 sin 7 sin 7 0 0.909 7. sec sec. csc 8 7.99 cos.. sin 8 0 7 8. cos 0 cos 0 0 00 sec 0 0.99 cos 0.00 9. cot.07 tan. 0. sec 8 0 sec 8 0 0 00.79 tan tan. 0.989 cos 8 0 cos 8 0 0 00 0.7. csc 0.87 sin.7. sec 9 0.9 tan 8 tan.7 0.987 cot 9 0 0.099
Section. Rigt Triangle Trigonometry 7. sin csc 0 0. cos tan º. sec cot 0. tan cos 0 0 7. csc sin 0 8. cot tan 0 sec cos 9. 0 0 tan 0 0 0 0 0. sin 0 y 8 y 8 sin 0 8 9. tan 0. sin 0 r r 0 0 0 sin 0. tan 8. tan tan 8 Heigt of te building: tan 8. meters Distance between friends: cos 8 y y cos 8 y 8 m Not drawn to scale 70 feet. meters. 000 ft sin 00 000 00 ft. tan tan w 00 w 00 tan 7. feet 0
8 Capter Trigonometry 7. ft y 0 ft sin 7. feet sin tan y y. feet tan Moving down te line: sin Dring vertically:.8 feet per second.7 feet per second 8. Let te eigt of te mountain. 9. Let te orizontal distance from were te 9 angle of elevation is sigted to te point at tat level directly below te mountain peak. Ten tan. and tan 9. tan 9 Substitute tan 9 tan 9 into te epression for tan.. tan. tan 9 tan. tan. tan 9 tan. tan 9 tan 9 tan. tan 9 tan. tan 9 tan. tan 9 tan..9 Te mountain is about. miles ig. tan 9 tan 9 0 0 sin 0 y y sin 0 8 cos 0 cos 0 8, y 8, 8 (, y) (, y) sin 0 y y sin 0 8 cos 0 cos 0 8, y 8, 8 70. tan tan d tan.7 centimeters
Section. Rigt Triangle Trigonometry 9 7. 0 (e) Angle, Heigt (in meters) 80 9.7 70 8.8 (f) Te eigt of te balloon decreases as decreases. 8 sin 8 0 0 sin 8 9.9 meters 0 7. 0. 0.9 0 0.0 0 (d) Te side of te triangle labeled will become sorter. 0.8 0. 7. 9., y. sin 0 y 0. 0 cos 0 0.9 0 cot 0.7 y sec 0 0.0 7. True, csc sin sin 0 csc 0 sin 0 sin 0 tan 0 y 0. csc 0 0 y.9 7. True, sec 0 csc 0 because sec90 csc. 7. False, 7. True, cot 0 csc 0 because cot csc cot csc cot csc. sin 0 cos 0 77. False, cot 0.7; sin 0 sin 0 sin 0.09 78. False, tan tan. tan tan 0. tan tan tan 0.0077 79. Tis is true because te corresponding sides of similar triangles are proportional. 80. Yes. Given tan, sec can be found from te identity tan sec. 8. In te interval 0. 0. 0. 0. 0. sin 0.0998 0.987 0.9 0.89 0.79 0, 0., > sin. As approaces 0, sin approaces. 8. sin 0 0.090 0.878 0.8090 0.9 cos 0.9 0.8090 0.878 0.090 0 0 8 7 90 CONTINUED
0 Capter Trigonometry 8. CONTINUED sin increases from 0 to as increases from 0 to 90. cos decreases from to 0 as increases from 0 to 90. (d) As te angle increases te lengt of te side osite te angle increases relative to te lengt of te ypotenuse and te lengt of te side acent to te angle decreases relative to te lengt of te ypotenuse. Tus te sine increases and te cosine decreases. 8., ± 8. t t t 9 t t t 9 t t t t 9 9 t t t t t t t t t t t t t t, t ±, 8. 0 0 0 8., 0, Section. Trigonometric Functions of Any Angle Know te Definitions of Trigonometric Functions of Any Angle. If is in standard position,, y a point on te terminal side and r y 0, ten: sin y r csc r y, y 0 cos r sec r, 0 tan y, 0 cot y, y 0 You sould know te signs of te trigonometric functions in eac quadrant. You sould know te trigonometric function values of te quadrant angles 0,,, and. You sould be able to find reference angles. You sould be able to evaluate trigonometric functions of any angle. (Use reference angles.) You sould know tat te period of sine and cosine is.