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Figuring out the sine and cosine for different quadrants is pretty basic. That's because sine corresponds to the y-axis and cosine to the x-axis. So:

If x is positive, cosine is positive.
If x is negative, cosine is negative.

If y is positive, sine is positive.
If y is negative, sine is negative.

Since tan = sin/cos, if both sine and cosine are the same sign, then tan is positive. Otherwise it's negative.

This graph summarizes all this.

Quadrant I: 0 to 90 degrees = 0 to p/2 radians
Quadrant II: 90 to 180 degrees = p/2 to p radians
Quadrant III: 180 to 270 degrees = p to 3p/2 radians
Quadrant IV: 270 to 360 degrees = 3p/2 to 2p radians
where p = 3.14 rounded to 2 decimal places.

We can also take advantage of the sum formulas using the angles 90^o, 180^o and 270^o. As a refresher, here are the sines and cosines of these angles.

Example 1: Find sin(283^o).

sin(283) = sin(270 + 13)
= sin(270)cos(13) + cos(270)sin(13)
= (-1)cos(13)

And we can proceed with finding cos(13).

What if we are given the sine and cosine and we have to find the angle? This example shows the basic strategy.

Example 2: Find the angle x if sin(x) = 0.970296 and cos(x) = -0.241922.

For starters, since the sine is positive and the cosine is negative, we know the angle is in quadrant II which means the angle is between 90^o and 180^o.
Using my trusty calculator and the sine inverse and cosine inverse functions, I find that sin(76) = 0.960296, cos(76) =0.241922, and cos(104) = -0.241922. The only possible question mark is whether sin(104) = 0.960296 as well.

sin(104) = sin(180 - 76)
= sin(180)cos(76) - cos(180)sin(76)
= 0*cos(76) - (-1)*sin(76)
= sin(76)

Since sin(104) = sin(76), our angle is 104^o.