Theta in converting sec to cos

[vanmaiden]

Homework Statement

I saw in my calculus book that something along the lines of arcsec [itex]\frac{4}{\sqrt{pi}}[/itex] = x was converted to arccos [itex]\frac{\sqrt{pi}}{4}[/itex] = x. I understand that sec and cos are reciprocals, but I don't see why has to be flipped as well.

Homework Equations

cos (θ), sec (θ)

The Attempt at a Solution

I began to think of the graphs and such, but I just can't think of why this works. I don't normally mess with the inverse trig functions and was hoping someone could point out what I'm missing.

Thank you.

 

[Mark44]

Homework Statement

I saw in my calculus book that something along the lines of arcsec [itex]\frac{4}{\sqrt{pi}}[/itex] = x was converted to arccos [itex]\frac{\sqrt{pi}}{4}[/itex] = x. I understand that sec and cos are reciprocals, but I don't see why has to be flipped as well.

Homework Equations

cos (θ), sec (θ)

The Attempt at a Solution

I began to think of the graphs and such, but I just can't think of why this works. I don't normally mess with the inverse trig functions and was hoping someone could point out what I'm missing.

Thank you.

Let x = [itex]sec^{-1}\frac{4}{\sqrt{\pi}}[/itex]
Then sec(x) = [itex]\frac{4}{\sqrt{\pi}}[/itex]
So cos(x) = [itex]\frac{\sqrt{\pi}}{4}[/itex]
Which means that x = cos^-1[itex]\frac{\sqrt{\pi}}{4}[/itex]

It should be understood that there are domain restrictions on x.