+0  
 
0
386
3
avatar

I was unable to make it to class when this was covered and I'm hoping to see a step-by-step example of how one would write this out.

Guest Feb 21, 2015

Best Answer 

 #2
avatar+94105 
+13

I always draw a quick pic for these.

let 

$$\\\theta=sec^{-1}u\qquad so \\
u=sec\;\theta\\
sec\;\theta = \frac{1}{cos\theta}=\frac{hyp}{adj}=\frac{u}{1}$$

 

Now I can read the positive answer straight off the triangle. 

 

$$cot(sec^{-1}(u))=cot\;\theta = \frac{adj}{opp}=\frac{1}{\sqrt{u^2-1}}$$

 

NOW u is positive so sec(theta) and cos(theta) are positive. So theta is in the 1st or 4th quadrant.

So cot(theta) can be positive or negative so

 

$$cot(sec^{-1}(u))=\pm\;\frac{1}{\sqrt{u^2-1}}$$

Melody  Feb 22, 2015
 #1
avatar+92674 
+8

The arcsec u just means an angle, Θ, whose secant = u.... or just .... u /1

Thus, since the secant ratio is r / x....this means that r = u   and x = 1

And y = √[r^2 - x^2]  = √[u^2 - 1^2] = √[u^2 - 1]

And the cotangent of such an angle is just x / y  =  1 / √[u^2 - 1]    and "rationalizing" the denominator, we have

cot (arcsec u) = cot Θ  = √[u^2 - 1] / [u^2 - 1]

 

CPhill  Feb 21, 2015
 #2
avatar+94105 
+13
Best Answer

I always draw a quick pic for these.

let 

$$\\\theta=sec^{-1}u\qquad so \\
u=sec\;\theta\\
sec\;\theta = \frac{1}{cos\theta}=\frac{hyp}{adj}=\frac{u}{1}$$

 

Now I can read the positive answer straight off the triangle. 

 

$$cot(sec^{-1}(u))=cot\;\theta = \frac{adj}{opp}=\frac{1}{\sqrt{u^2-1}}$$

 

NOW u is positive so sec(theta) and cos(theta) are positive. So theta is in the 1st or 4th quadrant.

So cot(theta) can be positive or negative so

 

$$cot(sec^{-1}(u))=\pm\;\frac{1}{\sqrt{u^2-1}}$$

Melody  Feb 22, 2015
 #3
avatar+92674 
0

Ah..thanks, Melody....I forgot that this angle could also be in the 4th quad, so I should have added that " -" to my answer...!!!

 

CPhill  Feb 22, 2015

5 Online Users

New Privacy Policy

We use cookies to personalise content and advertisements and to analyse access to our website. Furthermore, our partners for online advertising receive information about your use of our website.
For more information: our cookie policy and privacy policy.