Question

I had this question on my test.

If \(\cot \theta + \tan \theta = x\)

And \(\sec \theta - \cos \theta = y\)

Then evaluate \(\left( x^2 y + xy^2\right)^{2/3}\)

Also, could you guys just show me the solution, as I wouldn't be able to respond you with equations? A hint would also be fine, but it should be enough for me to know how to solve the whole problem. Thanks!

Answer 1

cot x= 1/tanx and sec x = 1/cos x

Answer 2

Aware of that. Let me try the problem again though, just to be sure I need you guys.

Answer 3

ok if a post a solution then wud it be prob?

Answer 4

write in terms of cos and sin ..will get answer pretty fast

Answer 5

\[\LARGE x=\frac{\cos^2 \theta+\sin^2 \theta}{2\sin \frac{\theta}{2}}=\frac{1}{2} cosec \frac{\theta}{2}\]
\[\LARGE y=\frac{1-\cos^2 \theta}{\cos \theta}=\frac{\sin^2 \theta}{\cos \theta}=\tan \theta \sec \theta\]
\[(x^2y+xy^2)^\frac{2}{3}=(\frac{1}{4}cosec^2 \frac{\theta}{2} \tan \theta \sec \theta+\frac{1}{2}cosec \frac{\theta}{2}\tan^2 \theta \sec^2 \theta)^\frac{2}{3}\]
hmmm..

Answer 6

How should I work toward my answer? I tried that but didn't seem to work...

Answer 7

Huh

Answer 8

lol

Answer 9

Wut

Answer 10

I think it is plain substitution

Answer 11

Tuw

Answer 12

Wouldn't that make it lengthy?

Answer 13

maybe that's the purpose then to evaluate by simplification

Answer 14

Yeah, but what to simplify?

Answer 15

how much simplified form do u want? i did it :|

Answer 16

The answer is a number

Answer 17

:3

Answer 18

Wait, did you do the complete answer or left the rest to me?

Answer 19

left the rest to you

Answer 20

A problem is that we haven't done half angle identities... only the three identities.

Answer 21

xy(x+y)
secx cosecx tanx sec x (secxcosecx+tanxsecx)
sec^3x( secxcosecx+tanxsecx)
\[\frac{1}{\cos x \sin x} (\frac{cosx}{\sin x \cos^3 x}+ \frac{\sin x \cos x}{\sin x \cos^3x})=\]
\[\Large \frac{1}{\sin x}(\frac{1+\sin x}{\sin x \cos^3x})=\frac{1+\sin x}{\sin^2x \cos^3 x}\]
\[\Large (\frac{1+\sin x}{\sin^2x \cos^3 x})^\frac{2}{3}\]=?

Answer 22

alternatively
((2 sin(theta)+cos(2 theta)-3)/(sin(2 theta)-2 cos(theta)))^(2/3)

Answer 23

WF result^ :P

Answer 24

I don't think we can get a numerical answer from this question anyway

Answer 25

if he said it is a number then it is a number.

Answer 26

ya it is a lot of work, so I just plugged it in wolfram

Answer 27

lazy in the AM syndrome

Answer 28

wolfram is awesome, but what happens if they have to do it manually meaning no computers?

Answer 29

screw manual then
I do it in class all the time

Answer 30

if you have to do manually then do easier one like mine :D

Answer 31

what about on an exam? I have some professors who don't allow calculators.

Answer 32

2/2 = 1

Answer 33

we have a winner here!

Answer 34

yeah. it is easy, just tedious

Answer 35

yeah I don't like tedious problems. I'm trying to memorize formulas here. better now than later since I'll be using them in future sections

Answer 36

I had some problems where I did all the computation for a one or a zero as an answer

Answer 37

this wil happen many times in your life :) especially in limits

Answer 38

yay a medal!

Answer 39

in the other thread...didn't realize that