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Basics of Trigonometry


Date: 12/12/2001 at 13:12:19
From: Coley 
Subject: Why is trig important?

Why is trigonometry important?

It is hard. Why is it necessary for us to know trig?


Date: 12/12/2001 at 16:18:46
From: Doctor Jeremiah
Subject: Re: Why is trig important?

Hi Coley,

It really isn't hard. If it seems hard that's because you were taught 
in a manner that doesn't work for you. Different people learn 
different ways, but teachers sometimes only teach one way (whatever 
way they are best at), whereas the way you are taught needs to be the 
way you learn.

One of the main things that needs to be taught when you learn 
something is what it's good for and what you need it for. If that 
wasn't part of what you learned, then that is probably why you don't 
understand it.

Trigonometry is necessary any time there is an angle involved (the 
angle of the sun, the angle of the cue ball in a game of pool...) or 
any time you have to compare two non-parallel lines (the height of a 
building and the height of its shadow, the distance a train is from
the crossing and the distance a car is from the crossing). These kinds 
of problems cannot easily be solved without trig.

But what is trig? There are only three important definitions in 
trigonometry. Everything else is based on those three. If you can 
understand these three definitions, you should have no trouble 
understanding any other part of trig.

First, consider a triangle with a right (90-degree) angle:

Pick an angle at random (not the 90-degree angle) and label it a, as I 
did below. Then label the side that doesn't touch the right angle as 
the hypotenuse (for the origin and meaning of the word hypotenuse, 
see: 

   Origin, Meaning of 'Hypotenuse' 
   http://mathforum.org/library/drmath/view/57748.html

Now one of the remaining sides is touching your angle. That is the 
adjacent side. And the other one? The other one is the side opposite 
to your angle. Now your triangle looks like this:

              +
             /|
            / |
           /  |
          /   |
         /    |
  hypotenuse  |
       /      |
      /    opposite
     /        |
    /         |
   / a        |
  +-----------+
    adjacent

Now here are the three most important definitions:

 sine of the angle = opposite / hypotenuse

 cosine of the angle = adjacent / hypotenuse

 tangent of the angle = sine of the angle / cosine of the angle

And in your case (where the angle is labelled "a") it is:

 sin(a) = opposite / hypotenuse

 cos(a) = adjacent / hypotenuse

 tan(a) = sin(a) / cos(a) = opposite / adjacent

So, for example, say we had the very famous Pythagorean triangle with 
sides of 3, 4 and 5:

              +
             /|
            / |
           /  |
          /   |
         /    |
      5 /     |
       /      |
      /       | 4
     /        |
    /         |
   / a        |
  +-----------+
        3

What is the value of a? Well we have all three sides so we could use 
any one of those equations, but if we pick one at random (the one for 
the tangent)

 tan(a) = opposite / adjacent
 tan(a) =    4     /    3

Now we know that the tangent of the angle "a" is 4/3 but that doesn't 
answer our question. We need to know the angle itself and not the 
tangent.

Think about the square and the square root. These are opposite 
functions and they cancel each other out. What if we had a function 
that could undo the tangent function. Lets pretend it exists and call 
it the arctangent. Then:

 arctan( tan(a) ) = arctan( opposite / adjacent )
 arctan( tan(a) ) = arctan( 4/3 )

Since they cancel each other   arctan( tan(a) ) = a
we end up with this:

 arctan( tan(a) ) = arctan( 4/3 )
                a = arctan( 4/3 )
                a = 53.1301 degrees

Now we just need a button on our calculator that does the arctangent 
function. But wait, there is one! So we can figure out the angle when 
we are given the lengths of the sides.

But what if we want to calculate the length of one of the sides and we 
know the angle.

              +
             /|
            / |
           /20|
          /   |
         /    |
        /     | h
       /      |
      /       |
     /        |
    /         |
   /          |
  +-----------+
        10

For example imagine this is a building.  And the shadow is 10 feet 
long. And we know that at this time of year and at this location on 
the Earth the sun has an angle of 20 degrees from the perpendicular.  
What is the height of the building?

If we wanted to find "h" (the height) we would need an equation that 
involved an angle and the side opposite from it (10) and the side 
adjacent (h). We do not want an equation that includes the hypotenuse.

Well, isn't that just the equation for the tangent of an angle?

                tan(a) = opposite / adjacent
               tan(20) = 10 / h

Now, using a little algebra (and using * to mean multiplication):

                tan(a) = opposite / adjacent
               tan(20) = 10 / h
           h * tan(20) = h * 10 / h
           h * tan(20) = 10
 h * tan(20) / tan(20) = 10 / tan(20)
                     h = 10 / tan(20)
                     h = 10 / 0.3640
                     h = 27.4748 feet

Now we just need a button on our calculator that does the arctangent 
function. But wait, there is one! So we can calculate any part of a 
right triangle if we know any other two parts.

But what about a triangle that doesn't have a 90-degree angle? What 
can we do about that? Well, it turns out that any shape can be 
completely made of right triangles.  Consider this triangle:

            +
          + 120  +
        +             +
      +                    +
    + 45                   15   +
  +----------------------------------+

That triangle doesn't have any right angles at all. But it's the same 
as these two right triangles:

            +
          + |    +
        +   |         +
      +     |              +
    +       |                   +
  +---------+------------------------+

So we could do all the algebra and figure it all out, or we could take 
the shortcut and use the equations that other people figure out. The 
most famous one is the cosine law. To use the cosine law we need to
label our triangle. But we can't use hypotenuse so we can't label it 
the same way. So we will use letters instead:

            +
          +  C   +      a
     b  +             +
      +                    +
    + A                     B   +
  +----------------------------------+
                c

Notice that corresponding letters of the alphabet are the opposite 
sides to their angles. That is very important. So side c is opposite 
angle C.

The cosine law looks like this:

 c^2 = a^2 + b^2 - 2ab cos(C)

It looks sort of like the Pythagorean Theorem, doesn't it? Remember, 
you could do the algebra yourself and get this equation by breaking 
the triangle into two right triangles.

Notice that to use the cosine law all you need is two sides and the 
angle between them.

            +
          +      +      8
        +             +
      +                    +
    +                      20   +
  +----------------------------------+
                 10

Here is a triangle with no 90-degree angles, but we do have enough 
information to use the cosine law. Let's label our triangle so that 
c is the side that doesn't have a value:

            +
          +      +      b=8
    c   +             +
      +                    +
    +                    C=20   +
  +----------------------------------+
               a=10


         c^2 = a^2 + b^2 - 2ab cos(C)
         c^2 = 10^2 + 8^2 - 2*10*8*cos(20)
         c^2 = 100 + 64 - 160*cos(20)
         c^2 = 164 - 160*cos(20)
         c^2 = 164 - 160*0.9397
         c^2 = 13.6492
 sqrt( c^2 ) = sqrt( 13.6492 )      sqrt = square root
           c = 3.6945

So now we can calculate sides of triangles that don't have 90-degree 
angles. And we could do angles too, but only if we have the lengths of
all three sides.

            +
          +      +      6
    4   +             +
      +                    +
    +                           +
  +----------------------------------+
                 8

If we label this triangle so that the angle we are looking for is 
called C:

            +
          +  C   +      b=6
  a=4   +             +
      +                    +
    +                           +
  +----------------------------------+
               c=8

              c^2 = a^2 + b^2 - 2ab cos(C)
              8^2 = 4^2 + 6^2 - 2*4*6*cos(C)
               64 = 16 + 36 - 48*cos(C)
               64 = 52 - 48*cos(C)
          64 - 52 = -48*cos(C)
               12 = -48*cos(C)
           -12/48 = cos(C)
             -1/4 = cos(C)
   arccos( -1/4 ) = arccos( cos(C) )
   arccos( -1/4 ) = C
 104.4775 degrees = C

And thats all there really is to trig. Most of the rest of the stuff 
is just algebra. Some of the algebra gets messy, but the trig really 
never gets more complicated than this.

For more info on trig you should look at the Dr. Math FAQ:

   Trigonometry Formulas
   http://mathforum.org/dr.math/faq/formulas/faq.trig.html   

If you want to talk about this more or if you have a specific 
question, you can send it and we will try to help you figure it out.

- Doctor Jeremiah, The Math Forum
  http://mathforum.org/dr.math/   
    
Associated Topics:
High School Trigonometry

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