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### Sine, Cosine, Tangent; Four-meter Ladder

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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?
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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

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

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

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

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

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

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Associated Topics:
High School Trigonometry

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