+0  
 
0
470
5
avatar

Tan x=2 ,cos2x=?

Guest Oct 6, 2014

Best Answer 

 #3
avatar+91432 
+10

Great work Chris and Geno.

I'm going to hit it a different way.

 

$$\\\tan(\alpha+\beta)=\frac{tan\alpha+tan\beta}{1-tan\alpha tan\beta}\\\\
\tan(2x)=\frac{2tanx}{1-tan^2x}\\\\\\$$

 

so draw a right angled triangle and make one of the acute angles = 2x

 

the opposite side is $$2tanx=2*2=4$$

 

the adjacent side is $$1-tan^2x=1-2^2=1-4=-3$$    (the negative sign just means it will not be in the first quad )

 

Using pythagoras the hypotenuse will be

 

$$\\h=\sqrt{4^2+(-3)^2}\\
h=\sqrt{16+9}\\
h=\sqrt{25}\\
h=5\\$$

so

 

$$\\cos(2x)
=\frac{adj}{hyp}
=\frac{-3}{5}
=\;-0.6$$

Melody  Oct 7, 2014
Sort: 

5+0 Answers

 #1
avatar+17705 
+5

If  tan(x) = 2, the x = 63.435°.

cos(2x) = cos(2·63.435°) = cos(126.870°) = -0.6

This can also be done without a calculator, if you need that explanation, please repost.

geno3141  Oct 6, 2014
 #2
avatar+80850 
+10

If tan x = 2, then y / x = 2 / 1

Then, using, √(x2 + y2 ) = r 

√(22 + 12) = √5 = r

So sin x = y/r =  2/√5

And using

cos 2x = 1 - 2(sin x)2

cos 2x = 1 - 2(2/√5)2

cos 2x = 1 - 2 (4/5)

cos 2x = 1 - 8/5

cos 2x = -3/5 = -(.6)     .....as geno found   !!!!!!!

 

CPhill  Oct 6, 2014
 #3
avatar+91432 
+10
Best Answer

Great work Chris and Geno.

I'm going to hit it a different way.

 

$$\\\tan(\alpha+\beta)=\frac{tan\alpha+tan\beta}{1-tan\alpha tan\beta}\\\\
\tan(2x)=\frac{2tanx}{1-tan^2x}\\\\\\$$

 

so draw a right angled triangle and make one of the acute angles = 2x

 

the opposite side is $$2tanx=2*2=4$$

 

the adjacent side is $$1-tan^2x=1-2^2=1-4=-3$$    (the negative sign just means it will not be in the first quad )

 

Using pythagoras the hypotenuse will be

 

$$\\h=\sqrt{4^2+(-3)^2}\\
h=\sqrt{16+9}\\
h=\sqrt{25}\\
h=5\\$$

so

 

$$\\cos(2x)
=\frac{adj}{hyp}
=\frac{-3}{5}
=\;-0.6$$

Melody  Oct 7, 2014
 #4
avatar+26397 
+5

Here's yet another way:

 

tanx = 2 implies sinx = 2cosx  ...(1)

 

cos2x = cos2x - sin2x.     Using (1) this is  cos2x = -3cos2x  ...(2)

 

sin2x + cos2x = 1.     Using (1) this is 5cos2x = 1   which means that  cos2x = 1/5   ...(3)

 

Put (3) in (2) to get   cos2x = -3/5

Alan  Oct 7, 2014
 #5
avatar+91432 
0

Thanks,

It is great when we all weigh in like this.

I like Chris's and Alan's ways best but they are all good.   

Melody  Oct 7, 2014

24 Online Users

avatar
avatar
We use cookies to personalise content and ads, to provide social media features and to analyse our traffic. We also share information about your use of our site with our social media, advertising and analytics partners.  See details