# TrigonometrySolving a trig equation

#### Bushy

##### Member
I have $h = 1+0.6 \cos \left( \frac{\pi t}{2}\right)$ and need to find where $h=1.5$ for $t\in [0,4]$

The period of the function is 4 and I get solutions of

$\displaystyle 1.5 = 1+0.6 \cos \left( \frac{\pi t}{2}\right)$

$\displaystyle0.5 = 0.6 \cos \left( \frac{\pi t}{2}\right)$

$\displaystyle \frac{5}{6} = \cos \left( \frac{\pi t}{2}\right)$

$\displaystyle \cos^{-1}\frac{5}{6} = \frac{\pi t}{2}$

$\displaystyle t = \frac{2\times \cos^{-1}\frac{5}{6}}{\pi} \approx 0.373, 4-0.373$

Can anyone find a mistake ere?

#### Sudharaka

##### Well-known member
MHB Math Helper
I have $h = 1+0.6 \cos \left( \frac{\pi t}{2}\right)$ and need to find where $h=1.5$ for $t\in [0,4]$

The period of the function is 4 and I get solutions of

$\displaystyle 1.5 = 1+0.6 \cos \left( \frac{\pi t}{2}\right)$

$\displaystyle0.5 = 0.6 \cos \left( \frac{\pi t}{2}\right)$

$\displaystyle \frac{5}{6} = \cos \left( \frac{\pi t}{2}\right)$

$\displaystyle \cos^{-1}\frac{5}{6} = \frac{\pi t}{2}$

$\displaystyle t = \frac{2\times \cos^{-1}\frac{5}{6}}{\pi} \approx 0.373, 4-0.373$

Can anyone find a mistake ere?
Hi Bushy, $\frac{5}{6} = \cos \left( \frac{\pi t}{2}\right)$

$\Rightarrow\frac{\pi t}{2}=2n\pi\pm\cos^{-1}\left( \frac{5}{6}\right)$

$\Rightarrow t=4n\pm\frac{2\cos^{-1}\left( \frac{5}{6}\right)}{\pi}\mbox{ where }n\in\mathbb{Z}$

Since $$t\in[0,4]$$ the only solutions are,

$t=3.627141002645325\mbox{ and }t=0.37285899735467$

So your solutions are correct. Kind Regards,
Sudharaka.