Precalculus

9th Edition

ISBN: 9780321716835

Author: Michael Sullivan

Publisher: Addison Wesley

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Question

Chapter 7.2, Problem 30AYU

To determine

To calculate: The exact value of expression $csc[tan−1(−2)]$ .

Expert Solution & Answer

Answer to Problem 30AYU

The exact value is y1 $−52$ .

Explanation of Solution

Given information:

$csc[tan−1(−2)]$ .

Formula used:

Equation $y=tan−1x$ means $tany=x$ .

Theorem :- For any angle $θ$ , in standard position, let $P=(x,y)$ be the point on the terminal side of $θ$ , that is on the circle $x2+y2=r2$ . Then

$sinθ=yrcosθ=xrtanθ=yx,x≠0$ $cosecθ=ry,y≠0secθ=rx,x≠0cotθ=xy,y≠0$ .

Calculation:

Consider ,

$csc[tan−1(−2)]$

Let

$θ=tan−1(−2)tanθ=(−2)$

where $−π2<θ<π2$ .

Since $tanθ<0$ , it follows $−π2<θ≤0$ .

This implies $θ$ lies in quadrant IV.

Now,

$tanθ=yx=−21$

$=y=−2;x=1$

Using equation of circle we have,

$=x2+y2=r2=(1)2+(−2)2=r2=1+4=r2=5=r2=r=±5$

Thus we have,

$=x=1;y=−2;r=5$ .

Now,

$csc[tan−1(−2)]=cscθ=−52$ ( where $cscry$ )

Hence , the exact value of $csc[sin−1(−2)]$ is $−52$ .

Chapter 7.2, Problem 29AYU

Chapter 7.2, Problem 31AYU

Chapter 7 Solutions

Precalculus

Ch. 7.1 - Prob. 1AYU Ch. 7.1 - Prob. 2AYU Ch. 7.1 - Prob. 3AYU Ch. 7.1 - Prob. 4AYU Ch. 7.1 - Prob. 5AYU Ch. 7.1 - Prob. 6AYU Ch. 7.1 - y= sin 1 x means _____, where 1x1 and 2 y 2 .Ch. 7.1 - Prob. 8AYU Ch. 7.1 - Prob. 9AYU Ch. 7.1 - Prob. 10AYU

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The exact value of expression csc [ tan − 1 ( − 2 ) ] .