Find Reference Angle for 28π/6.

Question

The answer needs to be in radians. I turned it to degrees, then back to radians with my answer of 5π/6. That is incorrect.

Please explain how to do it all in radians and also how to covert from radians to degrees to find the angle and back to radian.

Thank you!

Answer 1

In degrees, one "lap" around the unit circle is 360°. In radians, one "lap" around the unit circle is 2π radians. So, if you go one time around, you will be back at the start, meaning you can either add or subtract 2π from any angle location, and still be at the same spot when you get done.

Your angle location (28π/6) is a fraction in "6th's". 2π is the same as 12π/6 so we can subtract 12π/6 from your angle and still be in the same place. This "reduction" helps make the problem easier:

So, now our problem is "What is the reference angle for 2π/3?"

The reference angle is the acute angle formed with the x-axis so it is the supplement to 2π/3 or π/3

If you want to convert back and forth between degrees and radians, use the ratio the π radians = 180°.

If you have an angle in radians, to convert it to degrees, multiply by (180°)/(π radians). If you have an angle in degrees, to convert it to radians, multiply by (π radians)/(180°)

Answer 2

Doing the problem by converting to degrees, doing the math in units you're more comfortable with, and then converting back to radians was a good instinct. But it's probably best to keep the whole problem in radians, (1) so you don't get into trouble with the math and more importantly (2) to get used to using radians.

The first thing to do is reduce the fraction:

28π/6=14π/3

This number is much bigger than 2π, a full circle around the origin. Those 2π's just bring us back to our starting point. So let's break it down into a set of 2π's plus a remainder. Since we've got a 3 in the denominator, let's express 2π as 6π/3 so that it's easier to see:

14π/3 = (6+6+2)π/3 = 2π + 2π + 2π/3

(14)

I always find it really helpful to draw the angle. Most people think in terms of degrees rather than radians when they're visualizing angles, so it's okay here to temporarily convert to degrees for the purposes of drawing.

2π/3 radians ×(360°/2π radians) = 120°

This tells us it's in quadrant II, since it's between 90° and 180° (or between π/2 and π).

(Wyzant won't let me draw a picture, so I will leave that up to you!)

Your original angle is 2π/3.

The reference angle is just the acute angle between the x-axis and the line, no matter what quadrant you're in. (Although I definitely recommend noting in your solution that the reference angle is in quadrant II.) We can see that its value is

π-2π/3 = π/3

Against my own advice, here's how to do it degrees:

Convert to degrees: 28π/6 radians × (360°/2π radians) = 840°

Subtract all the full circles: 840° - 360° - 360° = 120° (quadrant II)

Subtract from 180° to find the reference angle: 180° - 120° = 60°

Convert back to radians: 60° × (2π radians/360°) = 2π radians × (60°/360°) = 2π radians × (1/6) = π/3

(notice that in the above step, I did not find the decimal value of 60° × (2π radians/360°) = 1.047 and then try to figure out how many π that is. I mean, you can! But there are so many opportunities for things to go wrong that it's better to keep π in the equation.

Answer 3

28π/6 = 14π/3 = 2π + 2π + 2π/3

28π/6 is coterminal with 2π/3, an angle with terminal side in Quadrant 2 (when drawn in standard position).

Reference angle = the acute angle with vertex at (0,0) formed by the terminal side and the x-axis

= π - 2π/3 = π/3.

In degrees:

28π/6 = 14π/3 = (14π/3)(180°/π) = 840° = 360° + 360° + 120°

Reference angle = 180° - 120° = 60°