2.2 Game 2302-0320 brief trigonometry tutorial  (Page 4/17)

The square of the adjacent side is 9. The square of the opposite side is 16. The sum of the squares is 25, and the square root of 25 is5. Thus, the length of the hypotenuse is 5.

A 3-4-5 triangle

You have created a rather unique triangle. You have created a right triangle in which the sides are either equal to, or proportional to the integervalues 3, 4, and 5.

I chose this triangle on purpose for its simplicity. We will use it to investigate some aspects of trigonometry.

The sine and arcsine of an angle

You will often hear people talk about the sine of an angle or the cosine of an angle. Just what is the sine of an angle anyway?

Although the sine of an angle is based on very specific geometric considerations involving circles (see (External Link) ), for our purposes, the sine of an angle is simply a ratio between the lengths of two different sides of a righttriangle.

A ratio of two sides

For our purposes, we will say that the sine of an angle is equal to the ratio of the opposite side and the hypotenuse. Therefore, in the case of the 3-4-5 triangle , the sine of the angle at the origin is equal to 4/5 or 0.8.

If we know the lengths of the hypotenuse and the opposite side, we can compute the sine and use it to determine the valueof the angle. (We will do this later using the arcsine.)

Conversely, if we know the value of the angle but don't know the lengths of the hypotenuse and/or the opposite side, we can obtain the value of the sine of theangle using a scientific calculator (such as the Google calculator) or lookup table.

The arcsine (inverse sine) of an angle

The arcsine of an angle is the value of the angle having a given sine value. In other words, if you know the value of the sine of an unknown angle, you canuse a scientific calculator or lookup table to find the value of the angle.

For example, we know that the sine of the angle at the origin on your 3-4-5 triangle is4/5. From that, we can determine the value of the angle. However, we probably can't do this calculation in our heads so we will use the Google calculator to computethe value of the angle.

We can also write a JavaScript script to perform the calculation, which we will do shortly.

Getting the angle for a known sine value

If you have access to a protractor, use it to measure and record the angle at the origin on your triangle. Then create an html file containing the code shown in Listing 2 and open it in your browser.