How would you find the exact radian measure of 540° without using a calculator?

θ= s r θ Previously we learned that the radian measure of an angle is the ratio of arc length to radius…

180° = π radians …and that when the arc length is half of the circle, the angle is π radians. This is an important fact to remember.

Some students misremember pi radians as a full circle (maybe because they confuse the shape of a pie?) Pi radians is only half of a circle!

We can find the radian measure of many angles by looking at the multiples of pi. Here is 2 pi, 3 pi, and 4 pi. We could keep going. Or we can go the other direction. Here’s negative pi, neg 2 pi, neg 3 pi, and so forth.

Half of pi, π 2 One third, π 3 One fourth, π 4 One sixth, π 6 Use fractions of pi to find special angles in the first quadrant. Half of a half circle is 90°, so 90° is half of pi radians, or π/2. Half of that, π/4, is half of 90, or 45°. And so on. The most commonly used “special” angles are π/2, π/3, π/4, and π/6, and their multiples. These are the ones you’ll need as you study trigonometry. To find the special angles in other quadrants, use multiples of those fractions. You really only need two: multiples of pi/6 and multiples of pi/4.

Multiples of π 6 : 7π 6 4π 3 3π 2 5π 3 5π 6 11π 6 2π 3 π 6 π 2 π 3 π Here are multiples of pi/6. Notice that the fractions are written in reduced form.

Multiples of π 4 : 5π 4 3π 2 3π 4 7π 4 π 4 π 2 π 5π 4 3π 2 3π 4 7π 4 π 4 π 2 π Here are multiples of pi/4. Again, the fractions are written in reduced form.

π 2 2π 3 π 3 3π 4 90° π 4 5π 6 120° 60° 135° 45° π 6 150° 30° 0° 180° 360° 210° 330° 7π 6 11π 6 225° 315° 240° 300° 5π 4 270° 7π 4 4π 3 5π 3 Here is the final result: all of the special angles that you should pay attention to labeled here in both degrees and radians. If you memorize that 180 degrees is pi radians, you can figure out all the rest out by using fractions and multiples of pi. You can make a similar diagram for negative angles going in the clockwise direction. 3π 2

540° - 360° = 180° = π So, how do you find the exact radian measure of 540° without using a calculator? First, recognize that it’s more than a full circle, so subtract 360°. It may help to sketch a picture of the angle. Now do you recognize it? It’s pi!