Division of fractions as used in Trigonometry.

 

Imagine that you have a right triangle where side 'a' is 5 1/8 inches long and side 'b' is 11 3/16 inches long.  If you need to determine the angle A you must use the tangent.  Follow along as we determine the tangent using the mixed numbers that are given for the lengths of the two sides of the right triangle.

 

What is the definition of the tangent? (You better know this!)

 

             Opposite side

Tangent A = ---------------

             Adjacent side

 

For this triangle the opposite side of angle 'A' is side 'a' which is 5 1/8 inches long, and the adjacent side is side 'b' which has a length of 11 3/16 inches.

 

Insert these values into the definition of the tangent.

 

             5 1/8

Tangent A = ---------

             11 3/16

 

Now you have a fraction which has a mixed number in the numerator and another mixed number in the denominator.  Rather an ugly mess at this point in time.  We will simplify it.

 

Let's convert each mixed number into just a fraction, which will be an improper fraction.  There is nothing wrong with improper fractions, so we will use them.  First we shall look at 5 1/8 inches.  What should we use as a denominator for this length?  Since there is a 1/8 in it we should use a denominator of 8. To convert 5 1/8 into a fraction with a denominator of 8 determine how many 1/8s (eights) are in 5.  How many are there?  Well, 5 times 8 is 40 so there are 40/8 in the whole number 5.  How many extra eights are there in 5 1/8?  There is 1 extra 1/8 which must be added to the 40/8 to get the total number of eights in 5 1/8.  Since 40/8 + 1/8 is 41/8 there are 41/8 in 5 1/8. In straight English we say five and one eighth contains forty-one eighths.

 

Use the same logic to determine the number of sixteenths in 11 3/16.  Use a denominator of 16.  How many sixteenths are in 11.  The answer will be 11 times 16 or 176, so there are 176/16 in 11 to which you must add the extra sixteenths, which in this case is 3.  Therefore 176/16 and 3/16 is 179/16.

 

Look at the problem as a fraction divided as a fraction.

 

 

 

 

             5 1/8

Tangent A = ---------

             11 3/16

 

 

             (5)(8)      (1)

             ------  +  -----

                (8)      (8)

Tangent A = ------------------

             (11)(16)    (3)

             -------  +  ----

                 (16)   (16)

 

 

               (40)      (1)

             ------  +  -----

                (8)      (8)

Tangent A = ------------------

                (176)    (3)

             -------  +  ----

                 (16)   (16)

 

 

                   (41)

                   ----

                    (8)

Tangent A =     -----------

                  (179)

                  -----

                   (16)

 

Now that we have a fraction divided by a fraction it is time to remember that a fraction divided by a fraction is the same as inverting the fraction in the denominator and then multiplying it by the fraction in the numerator.  Lets do it!

 

                 (41)     (16)

Tangent A =     ------ * -----

                  (8)    (179)

 

Multiply the numerators together and then multiply the denominators together to get the next step.

 

                 (41)(16)

Tangent A =     ----------

                  (8)(179)

 

At this point in time it should be remembered that sometimes fractions can be reduced, as in this case.  Since the 8 will go into the 16 twice the 8 can be reduced to a 1 while the 16 gets reduced to a 2.  This is shown in the next step.

 

                 (41)(2)

Tangent A =     ----------

                  (1)(179)

 

Now you should do the actual multiplication to get the next section.

 

                 (82)

Tangent A =     -----

                (179)

 

The tangent has now been determined to be:

 

             82

Tangent A =  ----  which does not reduce any more.

             179

 

This can also be written as:

 

Tangent A = 82/179

 

This process does take a bit of time but as you do more and more of them it becomes quicker and easier.  It took me a lot longer to type this so everything lined up correctly than it will take for you to do a few dozen of these problems.

 

If you wish to know the tangent as a decimal, so you can use it with the trig tables to determine the measure of an angle, simply divide the numerator by the denominator.  That is, put the numerator inside the division box and place the denominator on the outside and to the left of the box. (You should already know that, but I have included this information in case you forgot.)

 

Tangent A = 0.458100558 which to 4 decimal places becomes

 

Tangent A = 0.4581 or shown in the usual manner

 

Tan A = 0.4581

 

If you did not understand the process involved here you need to read this page over and over until it makes sense to you.