0.5 Phy1020: brief trigonometry tutorial  (Page 17/20)

The curves

A cosine curve is plotted with vertical values relative to the top grid line. It extends from -360 degrees on the left to +360 degrees on the right.

A sine curve is plotted with vertical values relative to the bottom grid line. It also extends from -360 degrees on the left to +360 degrees on theright. (Note once again that Figure 16 was flipped horizontally to crate a mirror image.)

Return values for the Math.asin, Math.acos, and Math.atan methods

I told you earlier that the Math.asin method returns a value between -PI/2 and PI/2. However, I didn't tell you that the Math.acos method returns a value between 0 and PI, or that the Math.atan method returns a value between -PI/2 and PI/2. You now have enough informationto understand why this is true.

Smooth curves

If you examine the two curves that you have just plotted, you can surmise that the sine and cosine functions are smooth curves whose values range between-1 and +1 inclusive. For every possible value between -1 and +1, there is an angle in the range -PI/2 and PI/2 whose sine value matches that value. There isalso an angle in the range 0 and PI whose cosine value matches that value.

(Although you haven't plotted the curve for the tangent, a similar situation holds there also.)

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.