WHERE THE FUNCTIONS COME FROM

We call two triangles similar triangles if their angles are the same, regardless of the length of the sides. The three triangles pictured here are similar triangles.  The little box in one angle means that it is a 90 degree angle, or a right angle, making these triangles right triangles.



As you can see, the triangles do not have to have the same orientation.

The reason we talk about similar triangles is that the sides of similar triangles are in proportion to each other: in the triangles above, the ratio of the two shortest sides in the first triangle is the same as the ratio of the two shortest sides in the second and third triangles:

4.8        3       4.5
           -------- = ---- = ------- = 0.75
6.4        4       6.0

The key to the ratios is that the sides have to be in the same order: I used the shortest side of each triangle on the top of the ratio, and the second shortest side on the bottom. If I use the shortest side on top and the longest side on the bottom I get

4.8      3       4.5
            ------- = ---- = -------- = 0.60
8.0       5       7.5

Divide each of the fractions, and verify for yourself that each one gives the same answer.

A right triangle is one where there is one 90 degree angle.  One of the facts about triangles is that the sum of the three angles adds up to 180 degrees, so if one is 90 degrees, then the other two add up to 90 degrees, and they must therefore each be less than 90 degrees.  Notice in the triangle below and in all of the other triangles on this page that the longest side is opposite the 90 degree angle and the shortest side is opposite the smallest angle.

We are going to talk only about right triangles.  Mathematicians give each of the possible ratios of two sides in a right triangle a name.



In the diagram to the right, I am going to deal only with the angle . I can talk about the sides of the triangle by referring to the sides that makes up the angle , but one of the sides that make up the angle is already called the hypotenuse.  {The hypotenuse is always the longest side, and it is always opposite the 90 degree angle.}  So the other side that makes up the angle  is the adjacent side. Then, the remaining side is opposite the angle .  [, by the way, is the Greek a, and is called alpha.]

On the triangles 1, 2 and 3 I have identified the opposite, adjacent, and hypotenuse for a specific angle. Notice that we do not name opposite and adjacent for the right angle, only the acute angles, and only one acute angle at a time. Below the diagram, fill in the side that is the opposite, adjacent and hypotenuse for each triangle.






4.  opposite ______	5.  opposite ______	6.  opposite ______
adjacent ______	adjacent ______	adjacent ______
hypotenuse ______	hypotenuse ______	hypotenuse ______

 Answers here.   You do it first.  It does you no good to just look at the answers and say 'I would have gotten that right.'  If you really already know this stuff, then doing the exercises won't take that much time and energy.
 

WHAT THE FUNCTIONS ARE


Now all we have to do is define our ratios. There are six possible ratios that we can form from the three sides. However, in our optics theory classes we will only use three of the the ratios, so we are only going to learn those three.

On your calculator there are three keys, and they are always in the same order on all calculators that I have seen here in the USofA. If you have one that does not have them in this order, let me know. The three keys are:


They are 'sine' (pronounced like sign) cosine and tangent.
The sine is defined as the ratio of the opposite to the hypotenuse.
The cosine is defined as the ratio of the adjacent to the hypotenuse.
The tangent is defined as the ratio of the opposite to the adjacent.


In other words,
            opposite
sine = -----------------
           hypotenuse

                adjacent
cosine = -----------------
              hypotenuse

                 opposite
tangent = ----------------
                 adjacent

or:



'Oscar Had A Heap Of Apples' is a fairly common mnemonic used to remember the order of the sides of the triangle for the order that the ratios are on the calculator.

Let's look at an example:


or:

 

Notice several rules:
 

Trigonometric Ratios The sine and cosine always have the hypotenuse in the denominator. Since the hypotenuse is the longest side, sine and cosine are never greater than 1. The tangent can be any number. It can be more than 1 or less than 1 or equal to 1. Since the sine, cosine, and tangent are ratios, they have no units. The sides must be in the same units before dividing.             opposite sine = -----------------            hypotenuse                 adjacent cosine = -----------------               hypotenuse                  opposite tangent = ----------------                  adjacent

OK, a few for you to do.






1.  sin a = _________	2.  sin b = _________	3.  sin c = _________
cos a = _________	cos b = _________	cos c = _________
tan a = _________	tan b = _________	tan c = _________

 Answers are here.  Do them yourself before you check them.

We will have one of two possible tasks to do in the formulas that we will learn.  You may be given an angle and asked to find its sine, cosine or tangent.  Or you may be given a sine, cosine or tangent and asked to find the angle.

Read the discussion on the types of calculators in the Optical Formulas Tutorial on page 6 (bottom) to 7 (top).  Decide if your calculator is type 'a' or type 'b'.

If you have type 'b'  skip to here.


TYPE 'a' CALCULATOR:

1. If you are told angle a = 50, and asked for sin a, punch 50, sin, into your calculator and you will get 0.7660.  That is the ratio of the opposite to the hypotenuse for a right triangle with an angle of 50 degrees.
2. If you asked for cos b and later told angle b = 36, punch 36, cos, into your calculator and you will get 0.8090.  That is the ratio of the adjacent to the hypotenuse for a right triangle with an angle of 36 degrees.
3. If you are asked for the tan 79, punch 79, tan, into your calculator and you will get 5.1446.  That is the ratio of the opposite to the adjacent for a right triangle with an angle of 79 degrees.

Your calculator will have one and only one of the following keys:  2nd, inv, shift.  When you see this: [2nd, inv, shift,] punch the one of these keys that you have.

1. If you are told that sin a = 0.50, punch 0.50, [2nd, inv, shift,], sin, and the calculator will say 30.  This is the angle that will have a ratio of opposite/hypotenuse = 0.50.
2. If the problem says cos b = 0.50, punch 0.50, [2nd, inv, shift,], cos, and the calculator will say 60.  An angle of 60 in a right triangle will have the ratio of adjacent side/hypotenuse = 0.50.
3. Tan c = 1.00.  What is angle c?  Punch 1.00, [2nd, inv, shift,], tan, and the calculator will say 45.  An angle of 45 in a right triangle will have the ratio of opposite side/adjacent side = 1.00.

 Skip type b instructions and go to exercises.

 
TYPE 'b' CALCULATOR:

1. If you are told angle a = 50, and asked for sin a, punch sin, 50, =, into your calculator and you will get 0.7660.  That is the ratio of the opposite to the hypotenuse for a right triangle with an angle of 50 degrees.
2. If you asked for cos b and later told angle b = 36, punch cos, 36, =, into your calculator and you will get 0.8090.  That is the ratio of the adjacent to the hypotenuse for a right triangle with an angle of 36 degrees.
3. If you are asked for the tan 79, punch tan, 79, =, into your calculator and you will get 5.1446.  That is the ratio of the opposite to the adjacent for a right triangle with an angle of 79 degrees.

 If this is not working for you, go here.

Your calculator will have one and only one of the following keys:  2nd, inv, shift.  When you see this: [2nd, inv, shift,] punch the one of these keys that you have.

1. If you are told that sin a = 0.50, punch [2nd, inv, shift,], sin, 0.50, =, and the calculator will say 30.  This is the angle that will have a ratio of opposite/hypotenuse = 0.50.
2. If the problem says cos b = 0.50, punch [2nd, inv, shift,], cos, 0.50, =, and the calculator will say 60.  An angle of 60 in a right triangle will have the ratio of adjacent side/hypotenuse = 0.50.
3. Tan c = 1.00.  What is angle c?  Punch [2nd, inv, shift,], tan, 1.00, =, and the calculator will say 45.  An angle of 45 in a right triangle will have the ratio of opposite side/adjacent side = 1.00.

Exercises:   [Round angles to whole angles.  Round ratios to four digits.]

1. cos  = 0.3333.  What is ? ______
2. tan  = 0.75.  What is ? ______ [That is beta, by the way, the Greek b.]
3. Angle r = 35 degrees.
1. What is the sin r?  ______
2. cos r = ______
3. tan r = ______
4. Angle i = 2.1 degrees.
1. sin i = ______
2. cos i = ______
3. tan i = ______
5. sin i = 0.6667.  i= ______
6. tan  = 1.50.  = ______
7. sin i = 1/1.66.  i = ______
8. sin r = 1.498/1.523.  r = ______

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