Q. 15 Explain why, if x聽=聽a聽sec聽u,... [FREE SOLUTION]

Chapter 5: Q. 15 (page 464)

Explain why, if $x = a s e c u$ , then $\sqrt{a^{2} s e c^{2} u 鈭� a^{2}}$ is $鈭� a \tan u$ if $x < 鈭� a$ and is $a \tan u$ if $x > a$ . Your explanation should include a discussion of domains and absolute values.

Short Answer

Expert verified

Ans:

$The absolute value sre significant since tanu is positive for u in 1^{st} quadrant and in 2^{st} quadrant its negative for u .$

Step by step solution

Step 1. Given information:

$\sqrt{a^{2} s e c^{2} u 鈭� a^{2}} x = a s e c u \{\begin{cases} 鈭� a \tan u & x < 鈭� a \\ a \tan u & x > a \end{cases}$

Step 2. Solving the trigonometric substitution:

Suppose $x 鈭� (- 鈭�, a] 鈭� [a, 鈭�), u 鈭� [0, \frac{蟺}{2}) 鈭� (\frac{蟺}{2}, 蟺]$ , and $x = a s e c u$

$\sqrt{x^{2} - a^{2}} = \sqrt{a^{2} s e c^{2} u - a^{2}} = \sqrt{a^{2} \tan^{2} u} = |a \tan u| \{\begin{cases} 鈭� a \tan u & , i f x < 鈭� a \\ a \tan u & , i f x > a \end{cases} T h e a b s o l u t e v a l u e s r e s i g n i f i c a n t \sin c e \tan u i s p o s i t i v e f o r u i n 1^{s t} q u a d r a n t a n d i n 2^{s t} q u a d r a n t i t s n e g a t i v e f o r u .$

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Explain why, if $x = a s e c u$ , then $\sqrt{a^{2} s e c^{2} u 鈭� a^{2}}$ is $鈭� a \tan u$ if $x < 鈭� a$ and is $a \tan u$ if $x > a$ . Your explanation should include a discussion of domains and absolute values.

Short Answer

Step by step solution

Step 1. Given information:

Step 2. Solving the trigonometric substitution:

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91影视

Explain why, if x=asecu, then a2sec2u鈭�a2is 鈭�atanuif x<鈭�aand is atanuif x>a. Your explanation should include a discussion of domains and absolute values.

Short Answer

Step by step solution

Step 1. Given information:

Step 2. Solving the trigonometric substitution:

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Geometry

Mechanics Maths

Statistics

Logic and Functions

Theoretical and Mathematical Physics

Applied Mathematics

Study anywhere. Anytime. Across all devices.

Explain why, if $x = a s e c u$ , then $\sqrt{a^{2} s e c^{2} u 鈭� a^{2}}$ is $鈭� a \tan u$ if $x < 鈭� a$ and is $a \tan u$ if $x > a$ . Your explanation should include a discussion of domains and absolute values.