Sine, Cosine, Tangent; Four-meter Ladder

Date: Sat, 15 Jun 96 11:34:31 -0700
From: Bill Hatcher
Subject: sin, cosine, tan

Dear Dr. Math,

Could you please explain how cosine, tangent and sin work? is 
there any way to perform these problems without a calculator? 

How would you find the sin, cosine, and tangent of this: 

A 4 meter ladder is placed against a wall at a 75 degree angle. 
How high up the wall does the ladder go?

Date: Sun, 16 Jun 1996 13:09:45 -0400 (EDT) 
From: Dr. Brian 
Subject: Re: sin, cosine, tan

Although there are uses for sine, cosine, and tangent for triangles 
other than right triangles, I will introduce them to you only in the 
context of right triangles. 

It shouldn't take too much convincing on your part that two right 
triangles whose non-right angles (the acute angles) are the same 
measurement are similar triangles. For instance, all triangles with 
angle combinations of 10, 80, and 90 degrees have the same basic 
shape, although the lengths of the sides may differ.

One of the properties of similar triangles is that the lengths of the 
corresponding sides always form identical ratios. That is, in our 
10-80-90 example, if two lengths of a smaller triangle are, etc.

Now since triangles with the same angles (similar triangles) have 
this similar ratio, someone (probably the Greeks, but I'm not sure) 
decided to express this ratio in terms of the specific non-right 
angles in the triangle. 

For one of the acute angles, we have these three names for the sides 
of the triangle:

   hypotenuse:   the longest side, across from the right angle. 

   adjacent leg: one of the two legs of the right triangle, and 
                 in this case, the one that forms part of the 
                 acute angle in question.

   opposite leg: the other leg, across from the acute angle. 

The sine, cosine, and tangent functions for an angle were then 
decided to be:

sine of the angle = ratio of opposite leg/hypotenuse 
cosine (angle) = adjacent leg/hypotenuse 
tangent (angle) = opposite/adjacent

Here's a quick example...Picture an equilateral triangle divided 
vertically down the middle. Each triangle formed is a right 
triangle, and they contain a 60 degree angle. 

     *
     **
     * *
     * *
     * *
     ******

The above diagram is what I'm talking about. Now for the 60 
degree angle on the left, the adjacent leg is the bottom of the 
triangle, and the hypotenuse is the longest side, on the left. Since 
the adjacent leg is half the length of the hypotenuse, we'd say that 
the COSINE of the 60 degree angle is 1/2.

* Typing cos 60 on a calculator should give you .5 * Different size 
angles will give you a different ratio from 0 to 1.


Now for your problem, you can use the same diagram I typed in 
earlier, except assume that the angle on the left is 75 degrees, 
not 60. The hypotenuse in this case represents your ladder, so its 
length = 4.

We need to know the height of the building, which would represent 
the OPPOSITE leg of the triangle. The ratio involving opposite 
with hypotenuse is SINE, so we have the equation:

sine of 75 degrees = (h, the height of the building)/(4, the length of 
the ladder)

Now, you *will* need a calculator or a table of values to figure out 
sine of 75 degrees. My calculator says 0.9659 

So 0.9659 = h/4, which solves to h = 3.86 meters. 

hope that helps

--Dr. Brian

Date: Sun, 16 Jun 1996 18:10:10 -0400 (EDT) 
From: Dr. Anthony 
Subject: Re: sin, cosine, tan

             A
            /|
           / |
    (Hypot/  |(Opposite)
         /   |
        /    |
       B---------- C
   (Adjacent)

In the right-angled triangle above, I shall be considering the trig. 
ratios of angle B, so B is called the 'reference' angle. 

Now in relation to B, the side AC is opposite B and is labelled the 
'Opposite' side. Also in relation to B, the side BC is next to or 
adjacent to B and is labelled the 'Adjacent' side. The side AB, 
opposite the right angle C is labelled the 'Hypotenuse' 

The hypotenuse is the longest side of the triangle; it is always the 
side opposite the right angle, whether B (as in this case) or A is the 
reference angle. If I had made A the reference angle then I should 
have had to relabel the sides BC and AC, but not the side AB. 

Now we define the trig. ratios as follows: 

sin(B) = AC/AB = OPP/HYP
cos(B) = BC/AB = ADJ/HYP
tan(B) = AC/BC = OPP/ADJ

Now, for any angle B, I can use a calculator or trig. tables to find 
the value of sin(B), cos(B) or tan(B). For example if B = 43 degrees, 
then 

sin(43) = 0.681998 cos(43) = 0.731354	tan(43) = 0.932515

If now I am given the length of one side, say AB, of the triangle, 
then I can calculate the other two sides. Suppose AB = 8 cms, then 
using the sine ratio, I have

AC/AB = sin(43)
AC = AB*sin(43)
AC = 5.455987

Using the cosine ratio:

BC/AB = cos(43)
BC = AB*cos(43)
= 5.85083

Now looking at your ladder problem, we are given that the length 
of the ladder is 4 metres (this corresponds to side AB), and the 
angle is 75 degrees (this corresponds to angle B). We are asked for 
the height up the wall that the ladder reaches, and this is the side 
AC. We shall therefore be using the sine ratio, to give

AC/AB = sin(75)
AC = AB*sin(75)
= 4*0.9659258
= 3.8637

So ladder reaches 3.8637 metres up the wall. 

-Doctor Anthony, The Math Forum