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sin 2 u 2 cos 2 u . sin u cos u
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EXAMPLE 8.3 Verify the identity tan u
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2 cot u
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We shall attempt to convert the right side of the possible identity into the left side by using the properties of fractions and substitutions using basic trigonometric relationships. sin 2 u 2 cos 2 u sin u cos u Thus, tan u 2 cot u sin 2 u sin u cos u 2 cos 2 u sin u cos u sin u cos u 2 cos u sin u tan u 2 cot u
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sin 2 u 2 cos 2 u . sin u cos u
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CHAPTER 8 Basic Relationships and Identities
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EXAMPLE 8.4 Verify the identity tan x
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csc x cos x . We transform the left side of the possible identity into a simpler form and then transform the right side into that same form. cot x tan x cot x csc x cos x Thus, tan x cot x sin x cos x cos x sin x sin 2x sin x cos x 1 # 1 sin x cos x cos 2 x sin x cos x 1 sin x cos x cos 2 x sin 2 x sin x cos x 1 sin x cos x
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1 csc x # cos x csc x cos x .
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EXAMPLE 8.5 Verify the identity
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sec x cot x tan x
sin x.
We transform the left side of the possible identity into an expression in sine and cosine and then simplify. sec x cot x tan x Thus, sec x cot x tan x 1 cos x cos x sin x cos x sin x sin x. 1 cos x cos x sin x # cos x sin x cos x sin x cos x sin x sin x cos 2 x sin 2 x sin x 1
sin x
EXAMPLE 8.6 Verify the identity
sin x cos x
cos x . sin x
We multiply the numerator and denominator on the left side by 1 cos x, which is the conjugate of the denominator. (The conjugate of a two-term expression is the expression determined when the sign between the two terms is replaced by its opposite.) The only time we use this procedure is when the product of the expression and its conjugate gives us a form of a Pythagorean relationship. 1 Thus, sin x cos x 1 1 1 cos x . sin x sec x A sec x tan x tan x 1 sec x tan x cos x # sin x cos x 1 cos x (1 1 cos x) sin x cos 2x (1 cos x) sin x sin 2x 1 cos x sin x
sin x cos x
EXAMPLE 8.7 Verify the identity
Since the left side has the radical, we want to multiply the numerator and denominator of the fraction under the radical by the conjugate of either. We will use the conjugate of the numerator since this will make the denominator the square of the value we want in the denominator. sec x A sec x tan x tan x sec x A sec x 1 sec x Thus, sec x A sec x tan x tan x 1 sec x tan x tan x . tan x # sec x tan x sec x tan x tan x sec 2 x A ( sec x tan 2 x tan x)2 1 A ( sec x tan x)2
Practice makes deciding which substitutions to make and which procedures to use much easier. The procedures used in Examples 8.3, 8.4, and 8.5 are the ones most frequently used.
(See Probs. 8.10 to 8.18.)
CHAPTER 8 Basic Relationships and Identities
SOLVED PROBLEMS
8.1 Prove the quotient relationships tan u
sin u and cot u cos u
y/x, and cot u x y x/r y/r
cos u . sin u
x/y, where P(x, y) is any point on the ter1 tan u cos u .R sin u
For any angle u, sin u y/r, cos u x/r, tan u minal side of u at a distance r from the origin. Then tan u y x y/r x/r sin u and cot u cos u
cos u .QAlso, cot u sin u
8.2 Prove the Pythagorean relationships (a) sin2 u (c) 1 cot2 u csc2 u.
For P(x, y) defined as in Prob. 8.1, we have A (a) Dividing A by r2, (x/r)2 (b) Dividing A by x , 1 Also, dividing sin2 u
cos2 u
y2 r2). 1.
1, (b) 1
tan2 u
sec2 u, and
(x2 tan u
(y/r)2
1 and sin2 u
cos2 u sin u 2 R cos u 1 Q
(y/x)
(r/x) and 1
sec2 u. 1 csc2 u. Q 1 2 R or 1 sin u cot2 u csc2 u Q 1 2 R or tan2 u cos u 1 sec2 u
cos2 u 1 cos2 u
1 by cos 2 u, Q (r/y)2 and cot2 u
(c) Dividing A by y2, (x/y)2 Also, dividing sin2 u
1 by sin 2 u, 1
cos u 2 R sin u
8.3 Express each of the other functions of u in terms of sin u.
cos2 u tan u sec u 1 sin2 u 21 1 21 sin 2u and sin u sin 2u cos u cot u csc u 21 21 1 tan u 1 sin u sin 2 u limits angle u to those quadrants (first and sin 2 u 21 sin 2 u sin u sin u cos u 1 cos u
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