# How do you find the amplitude, period and phase shift for y=cos(1/2(theta-90^circ))+2?

Jul 12, 2018

As below.

#### Explanation:

Standard form of the cosine function is
$y = A \cos \left(B x - C\right) + D$

Given $y = \cos \left(\frac{\theta}{2} - {\left(\frac{90}{2}\right)}^{\circ}\right) + 2$

$y = \cos \left(\frac{\theta}{2} - \frac{\pi}{4}\right) + 2$

$A = 1 , B = \frac{1}{2} , C = \frac{\pi}{4} , D = 2$

Amplitude $= | A | = 1$

Period $= \frac{2 \pi}{|} B | = \frac{2 \pi}{\frac{1}{2}} = 4 \pi$

Phase Shift $= - \left(\frac{C}{B}\right) = - \frac{\frac{\pi}{4}}{\frac{1}{2}} = - \frac{\pi}{2}$, $\frac{\pi}{2}$ to the LEFT

Vertical Shift $= D = 2$.