Download Chapter 11 Trigonometry

Chapter 11 Trigonometry
Sec. 1
Right Triangle Trigonometry
The most difficult part of Trigonometry is spelling it. Once we get by that, the rest is a piece of cake.
Before we start naming the trigonometric ratios, let’s look at a right triangle.
A
c
B
b
a
C
Using the sides; a, b, c, how many fractions can you make? Let’s see:
We have
a , b , a , then we have those fractions upside down; c , c , b . There are a total
c
c
b
a
b
a
of six different fractions. Since those fractions are comparing 2 sides of a triangle, we will call those
fractions RATIOS. So we have six different ratios. Now, how many trigonometric ratios
do you think there might be? If you answered six, you are right.
What we do is give those ratios names; we call them sine, cosine, tangent, cosecant, secant, and cotangent.
Now I don’t know who thought those names up, but we do have to know them.
Let’s look at that triangle again.
B
A
c
b
a
C
From your geometry, you might remember, the side opposite the right angle is called the hypotenuse. So
side c is always the hypotenuse, Now, if we stand on “A” a is opposite that angle A and b is adjacent to
angle A.
Now stay with me, if we stand on “B,” what side is opposite angle B? Yes, b. The hypotenuse does not
change, that is still c. What side is adjacent to angle B? All right, it is a.
The way we are going to name those trig ratios is in terms of where those sides are located with respect to
the angle we are working with (standing on).
If we are standing on A, what side is opposite A? If we are standing on B, what side is opposite B?
Notice the opposite side will be determined by where you are standing.
Now the way we name the trig ratios is by using the terms: hypotenuse, opposite, and adjacent. I want to
give three of those ratios we looked at names. The sine of an angle is defined to be the opposite side over
the hypotenuse. Symbolically stated:
SINE of an angle
= opposite side
hypotenuse
COSINE of an angle
= adjacent side
hypotenuse
TANGENT of an angle
=
opposite side
adjacent side
We typically abbreviate those names: sin, cos, and tan for sine, cosine, and tangent.
Now, we’ll look at another triangle using numbers instead of letters and find the sin, cos, and tan. You
will love this. Math is your life.
5
3
4
To help you remember the formulas for those three trig ratios, I will introduce you to SOHCAHTOA.
SOHOCAHTOA lives in a lean-to.
As you can see, his lean-to looks like a right triangle. We are not going to name his horse or what kind of
tree is in the picture. But SOHOCAHTOA’s name has some significance. Can you see it? OK, I’ll tell
you. S stands for sine, O for opposite, H for hypotenuse, A for adjacent, C for cosine, and T for tangent.
Therefore, Sine is the opposite over the Hypotenuse. Cosine is the Adjacent over the Hypotenuse, and the
Tangent is the Opposite over the Adjacent. Don’t you just love this?
Now, back to a triangle with numbers.
Y
5
X
3
Z
4
Let’s find the sin X, cos X, and the tan X. Remember SOHOCAHTOA.
sine X =
3
5
cos X =
4
5
tan X =
3
4
Was that hard? Of course not. Using the same triangle, find the sin Y, cos Y, and tan Y.
sine Y =
4
5
cos Y =
3
5
tan Y = ?
Make sure you spell “SOHOCAHTOA” correctly, that way you will know the trig ratios. As you can see,
this isn’t too bad. We’ll do more next time, but try this one for now.
R
Find the sin, cos, and tan for angles Q and R.
13
Q
Answer: sin R =
sin Q = 5
13
Sec 2.
12
, cos R =
13
, cos Q = 12
13
5
12
, tan R =
13
, tan Q = 5
12
5
S
5
12
Problem solving with trig ratios
Last time we looked at a right triangle, we found all the possible fractions that could be made by using the
sides of the triangle. We called those fractions ratios, then we gave those six ratios special names; sine.
cosine, tangent, cosecant, secant, and cotangent.
Now, the question that arises is what happens if we have a different size right triangle? We are going to
have to recall some geometry, in particular SIMILAR triangles.
Two triangles are SIMILAR if the corresponding angles are equal and all the corresponding sides are in
proportion. Yes, I am absolutely sure you remember that.
One theorem that I am also sure you will remember about similar triangles is: If a line is drawn parallel to
one of the sides of a triangle, then the two triangles formed are similar. That is the corresponding angles
are equal and the corresponding sides are in proportion.
Let’s look.
x
A
If I draw
y
B
C
XY anywhere so it is parallel to BC , then we have two triangles,
ABC and
AXY.
By using the previous theorem,
ABC is similar to AXY. That means angle A of the small triangle
equals angle A of the large triangle, angle X of the small triangle equals angle B, and by the same reasoning,
angle Y equals angle C.
It also means the sides are in proportion. That is, when I make the ratios for the small triangle using the sides,
those corresponding ratios for the large triangle will be the same.
EXAMPLE:
AY
AC
XY
BC
=
,
=
XY
BC
AX
AB
These ratios will NOT change unless the angle changes. The consistency allows to give these ratios the name
sine, cosine, tangent, etc.
Making that a little clearer; if the angle changes, then the ratio will also change.
Putting that more succinctly, the trigonometric measure of the angle of a triangle will have it’s own unique
ratio.
That means, the sine of a 30 degree angle, in a large or small triangle will be 1 . Remember the sine is the
2
opposite over the hypotenuse. SOHCAHTOA.
2
1
30°
3
That constancy will allow us to solve problems using trigonometry. By using many different right
triangles, we are able to find the ratios of the sides for different angles, We then are able to make what is
known as the TRIGONOMETRIC TABLES. Those tables allow us to look up those ratios given the
associated angles. Now that’s neat!
To use trig to solve problems, we have to know those trig ratios. I find it easy to remember them by
recalling SOCHCAHTOA.
EXAMPLE: A boy standing on level ground notices (or can measure) that the angle he has to look up
(angle of elevation) to see the top of a flagpole is 42 degrees. He can also measure the distance he is from
the pole and finds it to be 120 feet. How high is the flagpole?
SOLUTION: We’ll call the height of the flagpole “h.” Let’s look at a picture and fill in any
information we have
h
42°
120
Now, how can we determine the height? Well, since we are studying trig, let’s hope we use one of the trig
functions. Which one do we use? That’s the question.
The sine is the opposite over the hypotenuse; we don’t know the opposite or the hypotenuse. So we
won’t use the sine. Cosine is the adjacent over the hypotenuse; that does not give the height. So I won’t
use that. Tangent is the opposite over the adjacent. There, I know the adjacent is 120 and I’m looking for
the opposite (height). Therefore, I will use the tangent ratio.
I will have to look up the tan 42 on my trig table. The tan 42 = .9004
tan 42
=
.9004
=
opposite =
adjacent
h;
120
solving
h
120
120(.9004)
= h
Therefore the height of the flagpole is 108 feet.
Let’s examine what we know. All trig is in the study of the ratios of the sides of the right triangle. Those
ratios are given names; sine, cosine, tangent, cosecant, secant, and cotangent. To know which ratio we are
talking about, we stand on the vertex of the angle and use the acronym SOHCAHTOA. By looking at
different right triangles and knowing the ratios are the same for equal angles (using SIMILAR
TRIANGLES) we are able to make trig tables for all the angles that will give us the ratio of those sides.
Don’t you just love this stuff? You are probably thinking “ trig is my life.”