Question

Trig identity problem

Answer 1

Another one... I tried it, but :/

Answer 2

I simplified it to sin(4x)/(1 + cos4x) but I'm not sure if that's right... it might not be

Answer 3

@jim_thompson5910 up for another one?

Answer 4

ok still thinking, but I might have something

Answer 5

okay, thanks

Answer 6

Does (1 - cos4x)/(1 + cos4x) look familiar?

Answer 7

That's what I ended up with.

Answer 8

Cool! Do you know where to go from there?

Answer 9

Nope. Sorry.

Answer 10

Hah xD well, thanks for confirming my work anyway.

Answer 11

use the identity found on page 2 in the upper right hand corner ("Half Angle Formulas" section)

http://tutorial.math.lamar.edu/pdf/Trig_Cheat_Sheet.pdf

the identity
\[\Large \tan^2(\theta) = \frac{1-\cos(2\theta)}{1+\cos(2\theta)}\]

In this case, \[\Large \theta = 2x\]

Answer 12

Hmm how bout using conjugates?
Multiply top and bottom by the conjugate of your denominator,
it should turn your denominator into 1,
and then FOIL out the numerator.

Answer 13

This requires you to remember your third Pythagorean Identity though,\[\large\rm 1+\cot^2x=\csc^2x\]So if you feel more comfortable with sines and cosines then the other route is fine as well.

Answer 14

Omg you guys are lifesavers. Wish there were more of you around when the titanic sank.
Sorry... I'm tired.

Answer 15

Last one, I have 0 clue ..

Answer 16

`Wish there were more of you around when the titanic sank. ` I don't think I'd be of any help other than to calculate how big the iceberg is

Anyways, the first thing I'd do is use the unit circle

https://upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Unit_circle_angles_color.svg/2000px-Unit_circle_angles_color.svg.png

to determine the solutions to
\[\Large \cos(\theta) = \frac{1}{2}\]
those solutions are
\[\Large \theta = \frac{\pi}{3} + 2\pi*n\]
\[\Large \theta = \frac{5\pi}{3} + 2\pi*n\]
Do you agree so far?

Answer 17

Yes. Would the solutions for cos(6x) = 1/2 be different or the same?
And lol. Maybe you would have been about to calculate a way to fit everyone on the boats.

Answer 18

That's a good point.

So we have the solutions to theta. In this case, theta = 6x, so

6x = pi/3 + 2pi*n
or
6x = 5pi/3 + 2pi*n

Answer 19

to fully isolate x, multiply both sides by 1/6


Equation 1:
6x = pi/3 + 2pi*n
(1/6)*6x = (1/6)*[pi/3 + 2pi*n]
x = pi/18 + (pi*n)/3


Equation 2:
6x = 5pi/3 + 2pi*n
(1/6)*6x = (1/6)*[5pi/3 + 2pi*n]
x = 5pi/18 + (pi*n)/3

Answer 20

The general solutions to cos(6x) = 1/2 are

x = pi/18 + (pi*n)/3
or
x = 5pi/18 + (pi*n)/3

now plug in integer values of n to determine if x is in the interval from 0 to 2pi

2pi = 6.28 approximately

Answer 21

This question is kind of neat.
They didn't want the actual solutions, only the number of.
So I guess I was thinking of it in a different way...

We can produce 2 solutions for 6x...
what happens when you cut those 2 solutions each into 6 pieces?

Maybe my thinking is a little off, but it should work :DDD

Answer 22

Are there 12 solutions?

Answer 23

`Are there 12 solutions?`
yes in the interval [0,2pi]

Answer 24

Yeah!
Thank you guys so so much <3
Now I'm gonna go pass out...