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}.

Copyright © 2011 - Mathwizz.com all rights reserved