Evalulate the following, show reasoning.

[flaren5]

Hello all,

I am working on a question, but not sure how to show the reasoning.

The question is: "Evalulate the following, show reasoning."

sec^(-1)√(2)

just from assessing the question, I have come to the conclusion that the answer is π/4... (meaning pi/4).

Any insight on how to show reasoning for the evaluation would be greatly appreciated.
Thank you!

 

[DrPhil]

Hello all,

I am working on a question, but not sure how to show the reasoning.

The question is: "Evalulate the following, show reasoning."

sec^(-1)√(2)

just from assessing the question, I have come to the conclusion that the answer is π/4... (meaning pi/4).

Any insight on how to show reasoning for the evaluation would be greatly appreciated.
Thank you!

The inverse secant function can be read as "the angle whose secant is ...".

If the secant is sqrt(2), then the cosine is 1/sqrt(2)
and you should recognize what angle that is.
The principal value is the angle in the 1st quadrant - are you expected to find ALL angles whose cosines are 1/sqrt(2)?

 

[flaren5]

The inverse secant function can be read as "the angle whose secant is ...".

If the secant is sqrt(2), then the cosine is 1/sqrt(2)
and you should recognize what angle that is.
The principal value is the angle in the 1st quadrant - are you expected to find ALL angles whose cosines are 1/sqrt(2)?

Thank you, that does make things a little clearer. I don't believe that I am expected to find all the angles where the cosines are 1/sqrt(2).
I had thought it was in the first quadtrant, but wan't sure how to show this.

 

[DrPhil]

Thank you, that does make things a little clearer. I don't believe that I am expected to find all the angles where the cosines are 1/sqrt(2).
I had thought it was in the first quadtrant, but wan't sure how to show this.

Just giving the answer as pi/4 puts it in the first quadrant.

 

[lookagain]

If the secant is sqrt(2), then the cosine is 1/sqrt(2)
and you should recognize what angle that is.
The principal value is the angle in the 1st quadrant - are you expected to find ALL angles whose cosines are 1/sqrt(2)?

From the information in http://www.librow.com/articles/article-11/appendix-a-7, it looks as if the angle can only be \(\displaystyle \frac{\pi}{4}.\)\(\displaystyle \ \ \ \ \ And \ \ here: \ \ \)http://www.dummies.com/how-to/content/graph-inverse-secant-and-cosecant-functions.html

 

[HallsofIvy]

Thank you, that does make things a little clearer. I don't believe that I am expected to find all the angles where the cosines are 1/sqrt(2).
I had thought it was in the first quadtrant, but wan't sure how to show this.

I don't know what you mean by "show it was in the first quadrant". Either you were told to find an angle in the first quadrant or you were told to find all angles between 0 and \(\displaystyle 2\pi\) or you were told to find all such angles. Which was it?

 

[flaren5]

I don't know what you mean by "show it was in the first quadrant". Either you were told to find an angle in the first quadrant or you were told to find all angles between 0 and \(\displaystyle 2\pi\) or you were told to find all such angles. Which was it?

All that the question states is: "Evalutate the following, show reasoning."

I thought it was quite vague as well, hence why I'm here asking for anyones insight.
Thank you all your help.

 

[DrPhil]

From the information in http://www.librow.com/articles/article-11/appendix-a-7, it looks as if the angle can only be \(\displaystyle \frac{\pi}{4}.\)\(\displaystyle \ \ \ \ \ And \ \ here: \ \ \)http://www.dummies.com/how-to/content/graph-inverse-secant-and-cosecant-functions.html

When using a scientific calculator, only the principal value (1st or 2nd quadrant) is returned. However the function also exists in the 3rd and 4th quadrants. Since secant is an even function, the 4th quadrant is the same as the 1st: sec(-pi/4) = sec(pi/4). This problem is adequately solved by giving just the principal value.

 

[HallsofIvy]

Okay, I had misread the problem. \(\displaystyle sec(\theta)= \sqrt{2}\) for an infinite number of angle, \(\displaystyle \theta\). But a function can only have one value so to define the "inverse secant" function, we restrict the possible values of x in the secant to between \(\displaystyle 0\) and \(\displaystyle \pi\). \(\displaystyle sec^{-1}(\sqrt{2})\) must be in the first quadrant because \(\displaystyle \sqrt{2}\) is positive.