2.2 Game 2302-0320 brief trigonometry tutorial  (Page 15/17)

(Although you haven't plotted the curve for the tangent, a similar situation holds there also, but the maximum value is not limited to 1.0.)

An infinite number of angles

Therefore, given a specific numeric value between -1 and +1, there are an infinite number of angles whose sine and cosine values match thatnumeric value and the method has no way of distinguishing between them. Therefore, the Math.asin method returns the matching angle that is closest to zero and the Math.acos method returns the matching positive angle that is closest to zero.

What can we learn from this?

One important thing that we can learn is there is no difference between the sine or cosine of an angle and the sine or cosine of a different anglethat differs from the original angle by 360 degrees. Thus, the Math.asin and Math.acos methods cannot be used to distinguish between angles that differ by 360 degrees. (As you learned above, the situation involving the Math.asin and Math.acos methods is even more stringent than that.)

One-quarter cycle contains all of the information

Another thing that we can learn is that once you know the shape of the cosine curve from 0 degrees to 90 degrees, you have enough information to construct theentire cosine curve and the entire sine curve across any range of angles. Every possible value or the negative of every possible value that can occur in a sineor cosine curve occurs in the cosine curve between 0 degrees and 90 degrees. Furthermore, the order of those values is also well defined.

Think about these relationships

You should think about these kinds of relationships. As I mentioned earlier, as long as we are working with angles between 0 and 90 degrees, everything isrelatively straightforward. However, once we start working with angles between 90 degrees and 360 degrees (or greater), things become a little lessstraightforward.

If you have a good picture in your mind of the shape of the two curves between -360 degrees and +360 degrees, you may be able to avoid errors once you start working on physics problems that involve angles outsidethe range of 0 to 90 degrees.

Quadrants

We often think of a two-dimensional space with horizontal and vertical axes and the origin at the center in quadrants. Each quadrant is bounded byhalf the horizontal axis and half the vertical axis.

It is common practice to number the quadrants in counter-clockwise order with the upper-right quadrant beingquadrant 1, the upper-left quadrant being quadrant 2, the bottom-left quadrant being quadrant 3, and the bottom-right quadrant being quadrant 4.

Angles fall in quadrants

If you measure the angle between the positive horizontal axis and a line segment thatemanates from the origin, quadrant 1 contains angles between 0 and PI/2, quadrant 2 contains the angles between PI/2 and PI, quadrant 3 contains theangles between PI and 3*PI/2, and quadrant 4 contains the angles between 3*PI/2 and 2*PI (or zero). (Note that I didn't attempt to reconcile the inclusion ofeach axis in the two quadrants on either side of the axis.)