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
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 90o,
180o and 270o. As a refresher, here are the sines
and cosines of these angles.
Example 1: Find sin(283o).
sin(283) = sin(270 + 13)
= sin(270)cos(13) + cos(270)sin(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) =
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
90o and 180o.
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)
Since sin(104) = sin(76), our angle is 104o.