THE TANGENT AND OTHER TRIGONOMETRIC FUNCTIONS in Visual Studio .NET

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THE TANGENT AND OTHER TRIGONOMETRIC FUNCTIONS
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Fig. 28-3 28.6 The angle of inclination of a nonvertical line L is de ned to be the smaller counterclockwise angle from the positive x-axis to the line (see Fig. 28-4). Show that tan = m, where m is the slope of L .
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By taking a parallel line, we may always assume that L goes through the origin (see Fig. 28-5). L intersects the unit circle with center at the origin at the point P (cos , sin ). By de nition of slope, m= sin 0 sin = = tan cos 0 cos
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Fig. 28-4
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Fig. 28-5
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28.7 Find the angle at which the lines L1: y = 2x + 1 and L2: y = 3x + 5 intersect.
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Let 1 and 2 be the angles of inclination of L1 and L2 (see Fig. 28-6), and let m1 and m2 be the slopes of L1 and L2 . By Problem 28.6, tan 1 = m1 = 2 and tan 2 = m2 = 3. 2 1 is the angle of intersection. Now tan ( 2 1 ) = = = Since tan ( 2 1 ) = 1, 2 1 = radians = 45 4 tan 2 tan 1 1 + tan 1 tan 2 m2 m1 1 + m1 m2 [by Problem 28.3] [by Problem 28.6]
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5 5 3 2 = = =1 1 + ( 3)(2) 1 6 5
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In general, given tan ( 2 1 ), the value of 2 1 can be estimated from the table in Appendix D. It should be noted that, in certain cases, the above method will yield the larger angle of intersection.
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THE TANGENT AND OTHER TRIGONOMETRIC FUNCTIONS
Fig. 28-6
Supplementary Problems
28.8 Sketch the graphs of: (a) sec x; (b) cot x. tan u + tan v . 1 tan u tan v (b) Dx (tan x sec x) (e) Dx (sec2 3x) (h) Dx ( 3 tan x) (c) Dx (cot 2 x) (f ) Dx (csc(3x 5))
28.9 Prove the identity tan (u + v) =
28.10 Calculate:
x 5 2 (d) Dx (cot 4x + 3x) (g) Dx (csc x) (a) Dx 2 tan
28.11 Find y by implicit differentiation: (a) tan (xy) = y (c) tan2 (y + 1) = 3 sin x (b) sec2 y + csc2 x = 3 (d) y = tan2 (x + y) 3).
28.12 Find an equation of the tangent line to the curve y = tan x at the point ( /3,
28.13 Find an equation of the normal line to the curve y = 3 sec2 x at the point ( /6, 4). tan x x 0 x tan3 2x x 0 x 3 sin 3x x 0 tan 4x lim
28.14 Evaluate:
(a) lim
(b) lim
28.15 A rocket is rising straight up from the ground at a rate of 1000 kilometers per hour. An observer 2 kilometers from the launching site is photographing the rocket (see Fig. 28-7). How fast is the angle of the camera with the ground changing when the rocket is 1.5 kilometers above the ground 28.16 Find the angle of intersection of the lines L1: y = x 3 and L2: y = 5x + 4. 28.17 Find the angle of intersection of the tangent lines to the curves xy = 1 and y = x 3 at the common point (1, 1).
THE TANGENT AND OTHER TRIGONOMETRIC FUNCTIONS
[CHAP. 28
Fig. 28-7
28.18 Find the angle of intersection of the tangent lines to the curves y = cos x and y = sin 2x at the common point ( /6, 28.19 Find an equation of the tangent line to the curve 1 + 16x 2 y = tan (x 2y) at the point ( /4, 0). 28.20 Find the relative maxima and minima, in ection points, and vertical asymptotes of the graphs of the following functions, on [0, ]: (a) f (x) = 2x tan x (b) f (x) = tan x 4x 3/2).
28.21 Find the intervals where the function f (x) = tan x sin x is increasing. 28.22 GC Use Newton s method to approximate solutions of the following equations: on [ , 3 /2]; (c) tan x = 1/x on (0, ). 28.23 Evaluate lim 1 tan +x 1 . 4 (a) sec x = 4 on [0, /2]; (b) tan x x = 0
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