The German mathematician Karl Weierstrass (1815-1897) noticed that the substitution t = tan (a) If t = tan (₹). - < x < π, which of the right triangles confirms the following equalities? ³(²) COS (3) will convert any rational function of sin(x) and cos(x) into an ordinary rational function of t. 1 t (3) = √1+R² √1+t² and sin
The German mathematician Karl Weierstrass (1815-1897) noticed that the substitution t = tan (a) If t = tan (₹). - < x < π, which of the right triangles confirms the following equalities? ³(²) COS (3) will convert any rational function of sin(x) and cos(x) into an ordinary rational function of t. 1 t (3) = √1+R² √1+t² and sin
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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![(b) Show that cos(x)
cos(x) = cos 2
= 2
sin(x)
=
=
=
2
1-t²
1 + t²
Show that sin(x)
= sin 2
= 2
= 2
2t
t² + 1
(c) Show that dx
Since t = tan
=
=
X
2
1-t²
1 + t²
||
2
2t
1 + t²°
2
1 + t²
dt.
X
2
1
2
)² –
- 1
1
X
1
t² + 1
[by algebraic manipulation]
[by double angle formula cos(2x) = 2cos²(x) — 1]
[by part (a)]
[by simplification]
[by algebraic manipulation]
[by double angle formula sin(2x) =
=
[by part (a)]
[by simplification]
X =
Taking the derivative of both sides of this equality gives dx =
2
1 + t²
2sin(x)cos(x)]
dt, as was desired.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffab8b947-d70e-4af8-9844-5b96b790e388%2F984c996e-298d-4f2b-a802-0a0ad2a53a74%2Ffjjgb9m_processed.png&w=3840&q=75)
Transcribed Image Text:(b) Show that cos(x)
cos(x) = cos 2
= 2
sin(x)
=
=
=
2
1-t²
1 + t²
Show that sin(x)
= sin 2
= 2
= 2
2t
t² + 1
(c) Show that dx
Since t = tan
=
=
X
2
1-t²
1 + t²
||
2
2t
1 + t²°
2
1 + t²
dt.
X
2
1
2
)² –
- 1
1
X
1
t² + 1
[by algebraic manipulation]
[by double angle formula cos(2x) = 2cos²(x) — 1]
[by part (a)]
[by simplification]
[by algebraic manipulation]
[by double angle formula sin(2x) =
=
[by part (a)]
[by simplification]
X =
Taking the derivative of both sides of this equality gives dx =
2
1 + t²
2sin(x)cos(x)]
dt, as was desired.
Transcribed Image Text:The German mathematician Karl Weierstrass (1815-1897) noticed that the substitution t = tan
(a) If t = tan
(3).
-π < x < π, which of the right triangles confirms the following equalities?
2
1
cos (*)=√√²+² and sin(*)
COS
2
1
CHOICE A
O
X
CHOICE C
X
x/2
t² + 1
1
√√₁² +1
t
=
t
√ 1 + t²
t
1
CHOICE B
CHOICE D
X/2
NX
2
will convert any rational function of sin(x) and cos(x) into an ordinary rational function of t.
√√₁² +1
1
t² + 1
t
t
1
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