# Thread: sovle for x, use identity

1. ## sovle for x, use identity

i've figured out part of the problem but i dont' understand what part of it means and i don't know how to do it without a calculator

3 sin^2 x=cos^2 x 0<or equal to x <2pi - what does this inequality mean?

i know that sin^2 x + cos^2 x = 1 so i did this

3 sin^2 x = 1 - sin^2 x
but i don't know what how to solve for x from here and i can't use a calc

thanks!

2. ## hi

Hi!

$\displaystyle \mbox{Let} \ sin(x) \ = t$

Can you solve it from here?

3. Here's a slightly different method.

Simply divide by $\displaystyle \cos^2{x}$ on both sides and divide by 3 on both sides.

Domain: $\displaystyle 0\leq x <2pi$

Given: $\displaystyle 3\sin^2{x}= \cos^2{x}$

Divide by $\displaystyle \cos^2{x}$ on both sides: $\displaystyle 3\frac{\sin^2{x}}{\cos^2{x}} = 1$

Divide by 3 on both sides: $\displaystyle \frac{\sin^2{x}}{\cos^2{x}} = \frac{1}{3}$

Remember that:
$\displaystyle \tan{x} = \frac{\sin{x}}{\cos{x}}$

So let's replace now:

$\displaystyle \tan^2{x} = \frac{1}{3}$

Take the square root:

$\displaystyle \tan{x} = \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3}$

$\displaystyle x = \arctan{\left(\frac{\sqrt{3}}{3}\right)}$

To find the exact value without using a calculator, use the special 30-60-90 triangle. Check this out for more information:
Special right triangles - Wikipedia, the free encyclopedia

4. thanks everyone