Karl Weierstrauss (1815 – 1897) noticed that the substitution tan(;) t = will convert any rational function of sin x and cos x into an ordinary rational function of t. (a) If t= tan(G), -n

Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter6: The Trigonometric Functions
Section6.3: Trigonometric Functions Of Real Numbers
Problem 49E
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Karl Weierstrauss (1815 – 1897) noticed that the substitution
an()
t = tan
will convert any rational function of sin x and cos x into an ordinary rational
function of t.
(a) If t= tan(G, -n<x<t , sketch a right triangle or use
trigonometric identities to show that
and sin() - T+
1
t
cos
1 + t
(b) Show that
1
cos x =
2t
and sin x =
1 + t
1 + t?
2
(c) Show that dx
I+ ? dt.
1 + t?
Transcribed Image Text:Karl Weierstrauss (1815 – 1897) noticed that the substitution an() t = tan will convert any rational function of sin x and cos x into an ordinary rational function of t. (a) If t= tan(G, -n<x<t , sketch a right triangle or use trigonometric identities to show that and sin() - T+ 1 t cos 1 + t (b) Show that 1 cos x = 2t and sin x = 1 + t 1 + t? 2 (c) Show that dx I+ ? dt. 1 + t?
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