Question

Find the reference angle for -160 degrees?

Answer 1

`" "`
The Reference angle for `color(red)((-160^@)` is `color(blue)(20^@`

Explanation:

`" "`
Reference angle is the acute angle between the terminal side of the given angle and the horizontal axis.

It depends on the Quadrant the given angle belongs to, in degrees.

We will sketch the given angle in standard position.

Since the given angle is negative, we need to measure in the clockwise direction.

Since `color(red)(theta = (-160^@)`, we must determine the Coterminal angle.

Note that the given angle `color(red)(theta = (-160^@)` and the Coterminal angle add up to `color(blue)(360^@`

Hence, Coterminal angle will be `color(red)((-160^@+360^@)`

`rArr 200^@`

Now, our problem is reduced to finding the reference angle of the positive angle `200^@`

We can use the formula below, considering the Quadrant the angle lies in, to find the reference angle.

The new angle `color(red)(200^@` will line in `color(blue)("Quadrant III")`.

So we use the formula for Quadrant III to find the reference angle.

Hence, the required reference angle is `color(blue)(200^@ - 180^@=20^@`

The Reference angle for `color(red)((-160^@)` is `color(blue)(20^@`

I hope this helps.