Eric D. answered • 04/19/19
Math, Computer Science, and standardized testing tutor
First let's get the "greater than 360 degrees" part out of the way since that may sound intimidating but it really doesn't make the problem much harder.
In a unit circle there is a total of 360 degrees around. By convention, 0 degrees starts at coordinate (1,0) on the unit circle and increases when you travel counterclockwise on the circle.
So once you have for example, 361 degrees, where would you be on the unit circle? Well you would just be at the same place as you would be with 1 degree. Same is the case for (360*2)+1, and (360*3)+1 etc.
In general, any degree measure you have that's greater than 360 degrees would be on the same point on the unit circle as (measure % 360), a.k.a the remainder you get when you divide the measure by the number 360.
So continuing with the 361 degrees/1 degree example, how do we figure out the signs in the question?
You might know from trigonometry that the x coordinate on the unit circle corresponds to cos Φ and the y coordinate corresponds to sin Φ where Φ is the angle measure on the unit circle from the x axis. With 1 degree, we know that the x value is positive because it still lies in the quadrant (quadrant 1) to the right of the x axis. We also know the y coordinate is positive because 1 degree will put us directly above/counterclockwise from 0 degrees.
Since x and y coordinates are positive in this case, we also know that both cos Φ and sin Φ are positive. What about tan Φ? Well you might know that tan Φ = sinΦ/cosΦ. Since both sin and cos are positive, a positive number divided by a positive is also positive. So tan Φ is positive.
A similar analysis to this can be carried out for any angle measure above or below 360 degrees. Good luck, and please let me know if there are any follow up questions.
