# Let f(\theta)= 2\sin(\theta) - 1 and let g(\theta)= \frac{1}{2}\cos(\theta) + \frac{3}{2}. What...

## Question:

Let {eq}f(\theta)= 2\sin(\theta) - 1 {/eq} and let {eq}g(\theta)= \frac{1}{2}\cos(\theta) + \frac{3}{2}. {/eq} What is the midline and amplitude of g?

## Characteristics of the Cosine Function:

The characteristics that modify a Cosine function are the amplitude that modifies the maximum and minimum values of the function, the period that is the time that a cycle takes, the phase that determines the horizontal displacement of the function and the vertical displacement (midline).

{eq}\eqalign{ & {\text{The generic function of the cosine with }}\theta {\text{ - angle should have the following form:}} \cr & \,\,\,\,\,y = f\left( \theta \right) = A \cdot \cos \left( \theta \right) + B \cr & \,\,\,\,\,y = A \cdot \cos \left( {\left( {\frac{{2\pi }}{p}} \right)x + \phi } \right) + B,\,\,\,\,\,\,{\text{where:}} \cr & \theta :\,\,\,Angle \Rightarrow \theta = \left( {\frac{{2\pi }}{p}} \right)x + \phi \cr & A:\,\,\,Amplitude \cr & p:\,\,\,Period \cr & \phi :\,\,\,Phase\,\,Shift \cr & B:\,\,\,Vertical\,\,Shift\,\,\,\,\left( {Midline} \right) \cr & {\text{Then}}{\text{, in this particular case the function }}\,g\left( \theta \right) = \frac{1}{2}\cos \left( \theta \right) + \frac{3}{2},{\text{ by comparison}} \cr & {\text{we can conclude that:}} \cr & \,\,\,\,\boxed{A = \frac{1}{2}}\,\,\,\,{\text{ }}\left( {{\text{amplitude}}} \right) \cr & \,\,\,\,\boxed{B = \frac{3}{2}}\,\,\,\,\,{\text{ }}\left( {{\text{midline}}} \right) \cr} {/eq}