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1 tan u sec u tan u sec u 1 (NOTE: When expressed in terms of sin u and cos u, this identity becomes that of Prob. 8.17.)
SUPPLEMENTARY PROBLEMS
8.19 Find the values of the trigonometric functions of u, given sin u Ans. Quad I: 2/3, 25/3, 2/ 25 Quad II: 2/3, 8.20 25/3, 2 25/5, 25/2, 3/ 25 225/5, 25/2, 2/ 25 2/3. 3 25/5, 3/2 3/25 5/6. 6/5, 6/ 211 6/ 211 6 211/11 6 211/11 3 25/5, 3/2
Find the values of the trigonometric functions of u, given cos u Ans. Quad II: 211/6, Quad III: 5/6, 211/5, 5/ 211 5 211/11, 5 211>11, 211/6, 5/6, 211/5, 5/ 211
6/5, 5/4.
Find the values of the trigonometric functions of , given tan Ans. Quad I: 5/ 241 Quad III: 5> 241 5 241/41, 4/241 5241/41, 4/ 241
4 241/41, 5/4, 4/5, 241/4, 241/5 4 241/41, 5/4, 4/5, 241/4, 241/5
CHAPTER 8 Basic Relationships and Identities
Find the values of the trigonometric functions of , given cot Ans. Quad II: 1/2, 23/2, Quad IV: 1/2, 23/2, sin u sec u cos u csc u 1 23 1/ 23 23/3, 23, 23/3,
23. 2/ 23 23, 2 23 2 23/3, 2 2 23/3, 2
Find the value of Ans.
tan u when tan cot u
4/3.
Quad II: 23/5; Quad IV: 34/35
Verify the following identities.
8.24 8.26 8.28 8.30 8.32 8.34 8.36 8.38 8.40 8.42 8.44 8.45 8.46 8.47 sin (1 sin u csc u sec cos )(1 cos u sec u tan sec ) cot 1 1 sin 8.25 8.27 8.29 8.31 8.33 2 sec 2 A 8.35 8.37 8.39 8.41 8.43 (cot x y sin )2 sin2 )2 sin )2 r2 r2 cot y)(1 x2 y2 (1 sin2 A)(1 tan2 A) 1 cot A 1 tan u 1 1 (1 ( cot x 2 cos2 ) csc x)2 cot 1
csc2 x (1
cos2 x)
1 2 cos 2 A sin A cos A
tan A
tan2 x csc2 x cot2 x sin2 x 1 cos 2 u sin u 1 cos sin u 1 sin A sec
sin A cos A (tan A 1
cot A) sec u
sec u 1 1 tan sin x cot x sin 3u sin u
tan u
1 tan
1 sin A sin
cos x cos x csc
sec x sec x sec
sin u
sin u cos u csc x cot x 1
sec u sec u cos u sin x cos x csc u cot x cot y) (x cos r2(cos2 (r sin
tan x csc x cos 3u cos u
sin x tan x 1 sin u cos u tan u tan 2 u
sec x tan x cot u (tan x (x sin (2r sin (r sin
sin u cos u
sin u cos u cos 2 u sin 2 u tan x tan y) 0
tan y)(1 y cos )2 cos )2 cos )2
(r cos )2
Trigonometric Functions of Two Angles
9.1 Addition Formulas
sin ( cos ( tan (a ) ) b) sin cos cos cos cos sin sin sin
tan a tan b 1 tan a tan b
For a proof of these formulas, see Probs. 9.1 to 9.3.
9.2 Subtraction Formulas
sin ( cos ( tan (a ) ) b) sin cos 1 cos cos cos sin sin sin
tan a tan b tan a tan b
For a proof of these formulas, see Prob. 9.4.
9.3 Double-Angle Formulas
sin 2 cos 2 tan 2a 2 sin cos2 1 cos sin2 1 2 sin2 2 cos2 1
2 tan a tan 2a
For a proof of these formulas, see Prob. 9.14.
CHAPTER 9 Trigonometric Functions of Two Angles
9.4 Half-Angle Formulas
sin 2 u cos 2 u tan 2 u
1 1 1
1 A 1 A 1 A1
cos u 2 cos u 2 cos u cos u 1 sin u cos u 1 cos u sin u
For a proof of these formulas, see Prob. 9.15.
SOLVED PROBLEMS
9.1 Prove (1) sin ( and (2) cos (
Let and
sin cos cos cos
cos sin sin sin
when
are positive acute angles.
> 90 [Fig. 9.1(b)].
be positive acute angles such that
< 90 [Fig. 9.1(a)] and
Fig. 9.1
To construct these figures, place angle in standard position and then place angle with its vertex at O and its initial side along the terminal side of angle . Let P be any point on the terminal side of angle ( ). Draw PA perpendicular to OX, PB perpendicular to the terminal side of angle , BC perpendicular to OX, and BD perpendicular to AP. since corresponding sides (OA and AP, and OB and BP) are perpendicular. Then Now / APB sin (a b) AP OP AD OP cos a sin b AC OP OC OP sin a sin b DB OC OP DB OP OC # OB OB OP DB # BP BP OP DP CB OP DP CB OP DP OP CB # OB OB OP DP # BP BP OP
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