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.
Angle | sin | cos |
---|---|---|
0^{o} | 0 | 1 |
90^{o} | 1 | 0 |
180^{o} | 0 | -1 |
270^{o} | -1 | 0 |
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}.
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