Question

How do you find the amplitude, period, vertical and phase shift and graph `y=cos(theta-45)`?

Answer 1

First, you should rewrite the equation so it uses radians instead of degrees.
45° = `pi/4`

`y = cos (theta - pi/4)`

The standard form of a cosine function is `y=acos(bx + c) + d`, where `a` is the amplitude, `c` is the horizontal shift, and `d` is the vertical shift.

In this equation,
`a = 1`, `b = 1`, `c = -pi/4`, `d = 0`

The period of a cosine (and sine) function is `(2pi)/abs(b)`.
`(2pi)/abs(b) = (2pi)/abs(1) = 2pi`

Thus, the amplitude is 1, the period is `2pi`, there is no vertical shift, and there is a phase shift of `pi/4` to the right.

graph{cos (x-pi/4) [-10, 10, -5, 5]}