While out at the local amusement park with friends, you take a ride on the Go Karts. You ride around a circular track in the carts three and a half times, and then stop at a "pit stop" to rest. While waiting for your Go Kart to get more fuel, you are talking with your friends about the ride. You know that one way of measuring how far something has gone around a circle (or the trig values associated with it) is to use angles. However, you've gone more than one complete circle around the track.
Is it still possible to find out what the values of sine and cosine are for the change in angle you've made?
When you complete this Concept, you'll be able to answer this question by computing the trig values for angles greater than 360∘
The Unit Circle
Consider the angle 390∘. As you learned previously, you can think of this angle as a full 360 degree rotation, plus an additional 30 degrees. Therefore 390∘ is coterminal with 30∘. As you saw above with negative angles, this means that 390∘ has the same ordered pair as 30∘, and so it has the same trig values. For example,
In general, if an angle whose measure is greater than 360∘ has a reference angle of 30∘, 45∘, or 60∘, or if it is a quadrantal angle, we can find its ordered pair, and so we can find the values of any of the trig functions of the angle. Again, determine the reference angle first.
Find the value of the expression: sin420∘
420∘ is a full rotation of 360 degrees, plus an additional 60 degrees. Therefore the angle is coterminal with 60∘, and so it shares the same ordered pair, (12,3√2). The sine value is the y−coordinate.
Find the value of the expression: tan840∘
840∘ is two full rotations, or 720 degrees, plus an additional 120 degrees:
Therefore 840∘ is coterminal with 120∘, so the ordered pair is (−12,3√2). The tangent value can be found by the following:
Find the value of the expression: cos540∘
540∘ is a full rotation of 360 degrees, plus an additional 180 degrees. Therefore the angle is coterminal with 180∘, and the ordered pair is (-1, 0). So the cosine value is -1.
1. Find the value of the expression: sin570∘
2. Find the value of the expression: cos675∘
3. Find the value of the expression: sin480∘
1. Since 570∘ has the same terminal side as 210∘, sin570∘=sin210∘=−121=−12
2. Since 675∘ has the same terminal side as 315∘, cos675∘=cos315∘=2√21=2√2
3. Since 480∘ has the same terminal side as 120∘, sin480∘=sin120∘=3√21=3√2
Concept Problem Solution
Since you've gone around the track 3.5 times, the total angle you've traveled is 360∘×3.5=1260∘. However, as you learned in this unit, this is equivalent to 180∘. So you can use that value in your computations:
Find the value of each expression.
- Explain how to evaluate a trigonometric function for an angle greater than 360∘.