Question

How do you find the reference angle of `theta=-245^circ` and sketch the angle in standard position?

Answer 1

The reference angle is 65°.

Explanation:

Assuming an angle is in standard position, its reference angle is the smallest angle you can make between the terminal ray and the whole left-right `x`-axis.

Mathematically, the reference angle is the difference between the angle and the closest multiple of 180°.

`theta_"ref" = min{abs(theta-180n°):n in ZZ}`

An angle of `theta = –245°` lies in Quadrant II.
Thus, the closest multiple of 180° to `theta` is –180°.
The difference between these two is

`abs(–245° - (–180°))`
`=abs(–245°+180°)`
`=abs(–65°)`
`=65°`

Visual shortcut:

Draw the angle `theta`, then picture folding the plane in half, with the crease on the `x`-axis. Bend the two halves up. The reference angle is the angle that `theta` travels through if it were to fall towards the `x`-axis.