# Thread: Solving equation for x

1. ## Solving equation for x

1. arcsin(3x -pi) = 1/2

2. ## Re: Solving equation for x

Two things:

1) The argument for arcsin(x) and arccos(x) requires $\displaystyle -1 \le x \le 1$

3. ## Re: Solving equation for x

Originally Posted by TKHunny
Two things:

1) The argument for arcsin(x) and arccos(x) requires $\displaystyle -1 \le x \le 1$

1) Correct...I still don't understand it though. How can an arcsin form an angle of 1/2?

2) I posted the equations again below (in attachment). Little better format.

4. ## Re: Solving equation for x

1) Correct...I still don't understand it though. How can an arcsin form an angle of 1/2?

1/2 is an angle in radians
1. as mentioned earlier, note that ...

$\displaystyle -1 \le 3x-\pi \le 1$

therefore the value of x is restricted to ...

$\displaystyle \frac{\pi - 1}{3} \le x \le \frac{\pi + 1}{3}$

solving the equation ...

$\displaystyle 3x - \pi = \sin\left(\frac{1}{2}\right)$

$\displaystyle x = \frac{1}{3}\left[\sin\left(\frac{1}{2}\right) + \pi\right] \approx 1.207$

2. since arcsin(something) = arccos(something else) , the angle these two expressions equal can only be in quad I (why is that?)

let $\displaystyle \theta = \arcsin{\sqrt{2x}}$

$\displaystyle \cos{\theta} = \sqrt{1-2x}$

since the angles are equal ...

$\displaystyle \theta = \arccos{\sqrt{x}}$

$\displaystyle \cos{\theta} = \sqrt{x}$

so ...

$\displaystyle \sqrt{1-2x} = \sqrt{x}$

solve for $\displaystyle x$