Unit Circles and Special Angles (revised)?

Question

Ok so there is two types of special angles were working with the 30-60-90 Triangle and 45-45-90 triangle so my question is and were at the origin .So if i got 30 degrees how would i know how to find the value of x and y coordinates (is there anyway or will i have to remember the chart ?) like i see in my book that for a 45 degree angle the x= sq.rt of 2 and y= sq.rt of 2 will i have to remember these mentally or is there some way to calculate to get x and y from 45 degrees?

Answer 1

The x-value is equal to the cosine of the angle and the y-value is equal to the sine of the angle.

That said, it would be best to commit the values for 30, 45 and 60 degrees to memory. You will need them so much it will actually save time to have them memorized.
30 degrees: (sqrt(3)/2, 1/2)
45 degrees: (sqrt(2)/2, sqrt(2)/2)
60 degrees: (1/2, sqrt(3)/2)

Answer 2

This might help with memorization (you will need to remember which angles are in the table, though):

cos 0 = sqrt(4)/2 , sin 0 = sqrt(0)/2
cos 30 = sqrt(3)/2, sin 30 = sqrt(1)/2
cos 45 = sqrt(2)/2, sin 45 = sqrt(2)/2
cos 60 = sqrt(1)/2, sin 60 = sqrt(3)/2
cos 90 = sqrt(0)/2, sin 90 = sqrt(4)/2

As to getting values from the unit circle and geometry, this is easy for 45-45-90. Then the legs of the triangle are equal, and the hypotenuse is 1 (because it's a unit circle); so if A denotes the length of either leg, we have

A^2 + A^2 = 1, so 2A^2 = 1, so A = sqrt(1/2) = sqrt(2)/2.

So the (x,y) coordinates of the endpoint of the hypotenuse on the unit circle are (sqrt(2)/2, sqrt(2)/2), which are (cos(45), sin(45))