Angles and the Sine Function

Introduction: The Size of Angles

In this course you will need to understand a little bit about angles and the use of the Sine function. You already have an intuitive idea about angles. We deal with our intuitive understanding of angles in countless ways on a daily basis. What we need to do is develop a way to formally quantify this understanding. Consider an example; there are two angles shown below, and clearly the one on the left is a "bigger" angle than the one on the right.

The most common way that we specify angle size is by indicating how many degrees it is. For the two angles shown above, the angle on the left is approximately 30 degrees and the angle on the right is about 10 degrees. Since 30 degrees is more than 10 degrees, the angle on the left is bigger. That may seem obvious when you look at the two angles above, but sometimes the difference is small and not so obvious. You would probably have a hard time visually telling the difference between a 30 degree and a 31 degree angle.

The reason that we want to quantify the angle size is that it is sometimes necessary to be very accurate when dealing with technology.


The Sine Function

One angle size that is very important is 90 degrees. This is the angle between two lines that are perpendicular.

(We've also used the word normal to describe perpendicular lines.)

When a triangle has one angle that is 90 degrees it is called a right triangle. The triangle below is a right triangle.

The side labeled "b" is perpendicular ( 90 degrees ) to the side labeled "a".

We want to look at the angle q on the left side of the triangle between sides a and c. q is the Greek letter theta (pronounced they-ta) and it is often used to represent angles in mathematics. We could say how big it is by specifying the number of degrees like we did before. But there is another way that is used a lot in science and engineering. We can specify the angle in terms of the "proportions" of the triangle.

Suppose, for example, that we call the angle on the left side of the triangle q. Now let's introduce a new function:

Example: Suppose that in the triangle above, b = 3 and c = 5. Then sin q = 3/5 = .6

You can see that if sides b or c changed, then sin q would be different. That's why sin q is related to the shape, or sense of proportion, of the triangle. However, if all the sides of the triangle were made bigger or smaller proportionally, or in other words, the triangle were made bigger or smaller proportionally, then the angle q would still be the same.

 

Example: If the triangle were made twice as big, so that now a = 2 x 4 = 8, b = 2 x 3 = 6, and c = 2 x 5 = 10, then sin q = 6/10 = .6 This is the same answer as you got for the triangle on the left!

Intuitively, you can see that although the triangles are different sizes, their sense of proportion remains the same, and the angle q is the same size

The point of all this is that sin q is a measure of the angle. You can specify the size of the angle q by saying that q = 36.9 degrees (which it happens to be in this case), or by saying that sin q = .6. Both expressions mean the same thing! Another way to say it is that sin q will always be the same as long as q is the same. Being able to describe the measure of an angle q in terms of its degrees or in terms of sin q is similar to the way that we can describe a distance in terms of meters or yards.

If these two ways of measuring an angle are equivalent, then why bother using sin q in the first place? That's a fair question. For now, let's just say that there are many physical principles that are stated in terms of sin q and that it will be convenient for us to realize that and know how to work with it. Later in your studies you can find out more about the sin of an angle.


Using Your Calculator

For each angle between 0 degrees and 90 degrees, there is a corresponding sin q value. We could make a little table that relates the two values. A partial table is shown below:

q (degrees)

sin q

0

0

30

.5

45

.707

60

.866

90

1

What we really need to know is how to use your calculator to go back and forth between these two measures. (In the "old days" we carried around small booklets called "trig tables", or extensive tables were found in the backs of textbooks!) Fortunately, there is a button just for that purpose. The button will probably look like this:

Calculators vary a little from one model to another, so there are a couple of things you have to learn specifically for yours. Notice the use of two different colors on the "button" shown above. If you identify the correct button on your calculator you will see that there are indeed two different colors shown, although they may not be the same as the colors used here. You should also find on your calculator another button that is the same color as color of the printing above your sin button (the same color as the sin-1). The solid button may also say something like "shift", or "inv", or "second" on it. The reason is this: many buttons on scientific calculators have more than one function. You use the solid color button in a similar way to the way you use the shift key on a computer keyboard. In this case, you use:

sin to go from q in degrees to sin q

sin-1 (this nomenclature means "the angle whose sine is") to go from sin q to q in degrees

The first operation, knowing the angle and finding its sine, is like knowing an entry in the left column of the table above and needing the corresponding element in the right column.

The second operation, knowing the sine of an angle and finding the angle in degrees, is like knowing an entry in the right column of the table above and needing the corresponding element in the left column.

When you first turn on your calculator, it may just show the number 0 in the display window. Somewhere above the number you may see the notation "deg" or "rad". You want to have your calculator set so that it shows "deg". If it isn't, you will probably have to refer to your owners manual. Look for a procedure on setting degree mode. When you have done this you will be ready to continue. (Note: If you have a graphing calculator, the mode isn't displayed in the window. You need to press something like a "MODE" button to display this information.)

For what follows, realize that the exact sequence of buttons that you must press will vary slightly from one calculator model to another. Let's try a few examples:

Example 1: Find sin 45 degrees. Depending on the calculator, you might use either the first or second key sequence. (For example, my ancient Sharp Scientific calculator uses the first sequence, my TI 82 uses the second.)

You should get the Answer 0.7071

On some calculators you may have to change the sequence from those shown. You may or may not need to press the equal button, or the enter button on some calculators. Make sure you're able to get the correct answer before continuing.

Try one on your own:

Example 2: What is the sin of 36.9 degrees?

Answer: 0.6004

Now let's go the other direction- knowing a sine of an angle, find the corresponding angle in degrees.

Example 3: If sin q is equal to .4384, then what is q in degrees? Again, depending on the calculator, the sequence of buttons may be like either the top or the bottom example.

Answer: q = 26 degrees

Try one on your own.

Example 4: If sin q = 0.7997, what is q in degrees?

Answer: q = 53.1 degrees

A final word about nomenclature. When we go from sin q to q expressed in degrees as in the last two examples, the way this is usually expressed is like this: Sin-1 .4384 = 26 degrees. You say this as "the angle whose sine is .4384 is equal to 26 degrees." Don't confuse the "-1" in this context with the exponent that means "to the negative first power" (or, inverse). Older books may also have the notation "arc sine" which means the same thing.