Download Ch. 17 TrigR6

Chapter 15 Right Triangle 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
C
a
Using the sides; a, b, c, how many fractions can you make? Let’s see:
We have
a b b c c a
, , , , ,
c c a a b b
There are a total of six different fractions. Since those fractions are
comparing 2 sides of a triangle, we will call those 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.
A
c
b
B
a
C
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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 ∠A and b is adjacent to ∠A.
Now stay with me, if we stand on “B,” what side is opposite ∠B? Yes, b.
The hypotenuse does not change, that is still c. What side is adjacent to
∠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 is
opposite
hypotenuse
Cosine of an angle is
Tangent of an angle is
adjacent
hypotenuse
opposite
adjacent
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.
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To help you remember the formulas for those three trig ratios, I will
introduce you to SOHCAHTOA. SOHCAHTOA 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 SOHCAHTOA’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?
Example 1
Let’s find the sin B, cos B, and the tan B
A
5
B
3
4
C
Remember SOHCAHTOA.
sinB = 3/5
cosB = 4/5
tan B = 3/4
Was that hard? Of course not. Using the same triangle,
find the sin A, cosA, and tanA.
sinA = 4/5
cosA = 3/5
tanA = 4/3
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Make sure you spell “SOHCAHTOA” 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.
Example 2
Find the sin, cos, and tan for angles Q and R.
R
13
Q
5
S
12
Using SOHCAHTOA
sin Q = 5/13
cos Q = 12/13
tan Q = 5/12
sin R = 12/13
cos R = 5/15
tan R = 12/5
Just as a heads-up as you move forward in your study of trig, Look at the
relationships (ratios) of those angles. And know those relationships seem to
occur when the two angles are complementary.
Sec 2.
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.
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By definition, 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:
Theorem
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.
B
X
A
If I draw
XY
C
Y
anywhere so it is parallel to
triangles, ∆ABC and
BC
, then we have two
∆AXY.
By using the previous theorem, ∆ ABC is similar to ∆AXY. That means ∠A
of the small triangle equals angle ∠A of the large triangle, ∠X of the small
triangle equals ∠B, and by the same reasoning, ∠Y equals ∠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 2
Write 2 relationships based on the triangles being similar.
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.
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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
1/2. Remember the sine is the opposite over the hypotenuse. SOHCAHTOA.
2
1
30
A
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!
Use the tables in your calculator or in your book to find the sine, cosine and
tangent of the following angles.
1.
2.
3.
4.
30˚
60˚
40˚
63˚
Find the measue of the angle with the following values.
5.
6.
7
SinA = .5150
CosB = .7071
tanC = 1.3270
To use trig to solve problems, we have to know those trig ratios. I find it
easy to remember them by recalling SOCHCAHTOA.
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Example 3
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?
We’ll call the height of the flagpole “h.” Let’s look at a
picture and fill in any information we have
h
42
A
120
Now, how can we determine the height? Well, since we are
studying trig, let’s hope we use one of the trig ratios. 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 =
opposite
h
=
adjacent 120
.9004 =
h
120
so h is approximately 108 feet
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To solve problems involving the trigonometric ratios, it’s always wise to
think first, then set up a ratio. As we have said before, the choices we make
can either make easier or more cumbersome.
So, we have to ask, given certain information, am I going to use the sine
cosine or tangent and from what angle? So thinking first is important.
Example 4
Find the measure of DE in the diagram.
F
200
32
D
x
E
Using the trig ratios, we are working with 2 sides and an
angle. To solve problems, we need to know 2 of those 3
and find the unknown.
In this problem, we have 2 sides and an angle. Notice that
the hypotenuse is NOT one of the sides. For me, that
eliminates using the sine or cosine ratios.
So, that leaves me with the tangent ratio, do I want to use
angle D or F? Remember, if ∠D = 32˚, then ∠F = 58˚.
The sum must be 90˚
So now decide, which is an easier equation to solve, A or
B?
A. tan 32 =
200
x
or B. tan58 =
x
200
In either case, I have to use a table or calculator to find
the respective tangent values. However, since x is in the
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numerator of the B, I would choose to multiply, rather
then divide.
The tan 58˚ ≈ 1.6003,
tan58 =
x
200
x ≈ 200(1.6003) ≈ 320
In a number of applications, professionals use terms like “angle of
elevation” or “angle of depression”
B
angle of depression
angle of elevation
A
Line segment AB is the line of sight, the other two lines are horizontals.
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.”
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1.
The length of a kite string is 100 feet long. how high is the kite if the
kite make a 30˚ angle with the round?
2.
From the top of a building 300 feet high, you see a parked car on a
bridge, if the angle of depression is 60˚, how far is the car from the
building?
3.
A pilot reads the angle of depression from his position 5000 feet in the
air is 20˚, how far is the plane from the airport?
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