a To figure out if 7π3 radians gives an exact measurement, we should determine what rotation around the unit circle it corresponds to. Half a lap around the unit circle corresponds to an angle of 180^(∘) and an arc length of π. With this information, we can write the following equation.
180^(∘)=π
If we multiply this equation by 7 and divide by 3, we can determine the angle of rotation in degrees.
A 420^(∘) rotation corresponds to an arc length on the unit circle of 7π3.
To find some other angles that takes us to the same point on the circle, we have to either add or subtract 360^(∘) to our rotation. For example, if we subtract 360^(∘) from 420^(∘), we get a rotation of 60^(∘).
We can also add 2π, which takes us to an angle of 780^(∘).
We can rotate like this an arbitrary number of times, n. With this information, we can summarize all of the angles, in degrees, with a single expression.
60^(∘)± 360^(∘) n
b Let's draw the resulting right triangle.
c From Part A, we know that the reference angle to 420^(∘) is 60^(∘). Therefore, the right triangle we drew in Part B must be a 30^(∘)-60^(∘)-90^(∘) triangle. In such a triangle, the short leg is half the length of the hypotenuse and the long leg is sqrt(3) times greater than the short leg.
Now we can determine the sine and cosine value for the given distance around the unit circle.
sin (7Ï€/3)&= sqrt(3)/2 [1em]
cos (7Ï€/3)&= 1/2
To find the tangent value, we have to determine the ratio of the opposite side to the adjacent side.
