sin x 1 for x = 0 and ( /2) x ( /2). x in Visual Studio .NET

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sin x 1 for x = 0 and ( /2) x ( /2). x
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27.27 Determine whether the function f (x) = at x = /4
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sin x for x /4 is continuous at x = /4. Is the function differentiable cos x for x > /4
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GRAPHS AND DERIVATIVES OF SINE AND COSINE FUNCTIONS
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[CHAP. 27
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27.28 (a) The hypotenuse of a right triangle is known to be exactly 20 inches, and one of the acute angles is measured to be 30 , with a possible error of 2 . Use differentials to estimate the error in the computation of the side adjacent to the measured angle. (b) Use differentials to approximate cos 31 . 27.29 (a) Find the rst four derivatives of sin x. (b) Find the 70th derivative of sin x. 27.30 (a) Show that there is a unique solution of x 3 cos x = 0 in the interval [0, 1]. (b) GC Use Newton s method to approximate the solution in part (a). 27.31 GC Approximate by applying Newton s method to nd a solution of 1 + cos x = 0. 27.32 GC Use Newton s method to nd the unique positive solution of sin x = x/2.
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The Tangent and Other Trigonometric Functions
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Besides the sine and cosine functions, there are four other important trigonometric functions, all of them expressible in terms of sin x and cos x. sin x De nitions: Tangent function tan x = cos x 1 cos x = Cotangent function cot x = sin x tan x 1 Secant function sec x = cos x 1 Cosecant function csc x = sin x EXAMPLE Let us calculate, and collect in Table 28-1, some of the values of tan x.
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sin 0 0 = =0 cos 0 1 1 1 sin( /6) 1/2 3 3 = = = = tan = 6 cos( /6) 3 3/2 3 3 3 sin( /4) 2/2 tan = = =1 4 cos( /4) 2/2 3/2 sin( /3) = = 3 tan = 3 cos( /3) 1/2 tan 0 =
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Notice that tan ( /2) is not de ned, since sin ( /2) = 1 and cos ( /2) = 0. Moreover,
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x ( /2)
tan x =
x ( /2)
sin x = + cos x
x ( /2)+
tan x =
x ( /2)+
sin x = cos x
Copyright 2008, 1997, 1985 by The McGraw-Hill Companies, Inc. Click here for terms of use.
THE TANGENT AND OTHER TRIGONOMETRIC FUNCTIONS
[CHAP. 28
because cos x > 0 for x immediately to the left of /2 and cos x < 0 for x immediately to the right of /2. Table 28-1 x 0 6 4 3 tan x 0
3 0.58 3 1 3 1.73
Theorem 28.1: The tangent and cotangent functions are odd functions that are periodic, of period . That they are odd follows from Theorem 26.3, sin x sin ( x) = = tan x cos ( x) cos x 1 1 cot ( x) = = = cot x tan ( x) tan x
tan ( x) =
The periodicity of period follows from Problem 26.15(c) and (d), tan (x + ) = sin (x + ) sin x = = tan x cos (x + ) cos x Dx (tan x) = sec2 x Dx (cot x) = csc2 x Dx (sec x) = tan x sec x Dx (csc x) = cot x csc x For the proofs, see Problem 28.1. EXAMPLE From Theorem 28.2 and the power chain rule,
2 Dx (tan x) = Dx ((sec x)2 ) = 2(sec x)Dx (sec x)
Theorem 28.2 (Derivatives):
= 2(sec x)(tan x sec x) = 2 tan x sec2 x
2 Now in (0, /2), tan x > 0 (since both sin x and cos x are positive), making Dx (tan x) > 0. Thus (Theorem 23.1), the graph of y = tan x is concave upward on (0, /2). Knowing this, we can easily sketch the graph on (0, /2), and hence everywhere (see Fig. 28-1).
CHAP. 28]
THE TANGENT AND OTHER TRIGONOMETRIC FUNCTIONS
Fig. 28-1
Theorem 28.3 (Identities): tan2 x + 1 = sec2 x and cot 2 x + 1 = csc2 x. Proof: Divide sin2 x + cos2 x = 1 by cos2 x or sin2 x.
Traditional De nitions As was the case with sin , and cos , the supplementary trigonometric functions were originally de ned only for an acute angle of a right triangle. Referring to Fig. 26-3, we have
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