Question

Find the reference angle for 8 pi divided by 5?

Answer 1

Reference angle `= color(blue)(288^0)`

Explanation:

`pi^c = 180^0`

`:. (8pi) / 5 = (8 * 180) / 5 = (cancel(1440) color(blue)(288)) / cancel(5) = 288^0`

Answer 2

Reference angle of `color(blue)[(8Pi)/5 = (2Pi)/5]`

The angles are in Radian measure.

Relationship between Radian and Degrees:

`Pi`** radians `= 180` degrees.

Explanation:

Given:

Original angle = `color(brown)((8Pi)/5)`

Reference angle is the angle formed by the terminal ray of the original angle and the x-axis.

`color(green)(Step-1)`

Note that, if we go counter-clockwise half-way around, through the first and second quadrants, it is Positive and is equal to `color(blue)(Pi.`

If we go further, through the third quadrant, the angle is Positive and is equal to `color(blue)((3Pi)/2.`

If we go further, through the fourth quadrant and reach the point where we started, the angle is Positive and is equal to `color(blue)(2Pi.`

Refer to the graph template below with `(0, 2PI), Pi/2, Pi, (3Pi)/2` identified.

We will use this graph template to identify our original angle **`color(green)((8Pi)/5)`

`color(green)(Step-2)`

Our Original angle`color(green)((8Pi)/5)` will lie in the fourth quadrant.

Hence, our required Reference angle is `=2Pi-(8Pi)/5` and the reference angle we will find.

`2Pi-(8Pi)/5`

`rArr (2Pi)/1-(8Pi)/5`

`rArr [(5*2Pi)-(8Pi)]/5`

`rArr [(10Pi)-(8Pi)]/5`

`rArr (2Pi)/5`

Hence,

Reference angle of `color(blue)((8Pi)/5)` is equal to `color(blue)((2Pi)/5)`

Analyze the graph below to understand how a reference angle is found: