# Thread: x=pi/2 is a solution?

1. ## x=pi/2 is a solution?

Solve $\displaystyle 2/(\cot {x}) + 3/(\csc {x}) = 0$, $\displaystyle 0<x<2\pi$.

After a few steps, I obtained

$\displaystyle (\sin {x})(2+3 \cos [x}) = 0$

$\displaystyle \sin {x} = 0$ or $\displaystyle \cos {x} = -2/3$

From $\displaystyle \sin {x} = 0$, I obtained the solution $\displaystyle x=\pi/2$. Is $\displaystyle x=\pi/2$ is a solution?

2. ## Re: x=pi/2 is a solution?

Originally Posted by woo
Solve $\displaystyle 2/(\cot {x}) + 3/(\csc {x}) = 0$, $\displaystyle 0<x<2\pi$.

After a few steps, I obtained

$\displaystyle (\sin {x})(2+3 \cos [x}) = 0$

$\displaystyle \sin {x} = 0$ or $\displaystyle \cos {x} = -2/3$

From $\displaystyle \sin {x} = 0$, I obtained the solution $\displaystyle x=\pi/2$. Is $\displaystyle x=\pi/2$ is a solution?
Hints:
1. Look at the first term in the original equation.
2. What is csc(0)?
3. Is x = 0 permitted in the original equation?

-Dan

3. ## Re: x=pi/2 is a solution?

Originally Posted by woo
Solve $\displaystyle 2/(\cot {x}) + 3/(\csc {x}) = 0$, $\displaystyle 0<x<2\pi$.
After a few steps, I obtained $\displaystyle (\sin {x})(2+3 \cos [x}) = 0$
$\displaystyle \sin {x} = 0$ or $\displaystyle \cos {x} = -2/3$
From $\displaystyle \sin {x} = 0$, I obtained the solution $\displaystyle x=\pi/2$. Is $\displaystyle x=\pi/2$ is a solution?
Did you even bother to look at the graph?

4. ## Re: x=pi/2 is a solution?

$\sin{x} = 0 \implies x = k\pi$, which are not valid solutions.

the Wolfram graph does not show the point discontinuities on the x-axis at integer multiples of $\pi$

The valid solutions are where $x = \arccos\left(-\dfrac{2}{3}\right)$

5. ## Re: x=pi/2 is a solution?

Thanks a lot.