Download FT U6 L2

6.2
Right Triangle Trigonometry
Lesson Objectives:
Understand
what each of the three trigonometric ratios tell about the
relationship between a set of sides in a right triangle
Understand that each of the trigonometric ratios
can be thought of as a percent
Use these trigonometric ratios to find a missing
side in a right triangle
Use the inverse trigonometric ratios to find a
missing angle in a right triangle
Consider This…
The Americans with Disabilities Act (ADA), signed
into law in 1990, included an outline of minimum
specifications need to make a ramp safely
“handicapped accessible.” These specifications
included a stipulation that the largest possible
angle between the start of the ramp and the
ground be a mere 4.76o.
When remodeling a building, a contractor realizes
that she needs to replace part of a set of stairs
with a ramp. The top of the staircase is 2 feet
above the ground. How will the contractor know
how long she needs her ramp to be in order to
meet minimum ADA specifications?
2 ft
?
?
?
?
?
Consider This…
1. Use the triangles pictured below to help you fill in the table. The diagrams are drawn to
scale, with 1 cm : 1 ft. Using a ruler, measure the length of each “ramp” (that is, the
hypotenuse of each right triangle), and then using a protractor, approximate the measure
of the angle between the “ramp” and the “ground.”
h (hypotenuse)
A (angle, in degrees)
Scale Drawings of Possible Ramp Shapes:
A
E
B
H
C
F
M
O
D
I
G
K
N
L
J
Q
R
P
2. In all of the above triangles, what is the length (in cm) of the side opposite the “ramp-toground” angle?
3. Write a statement describing the relationship between the length of the hypotenuse and
the measure of the “ramp-to-ground” angle.
4. If you were the contractor, which of the above options would you choose, and why?
5. For your option, write a ratio of the length of the side opposite the “ramp-to-ground” angle
to the length of the ramp (hypotenuse). Then write a ratio of the length of the side
opposite the “ramp-to-ground” angle to the horizontal length of the ramp on the “ground.”
Summarize
The “ramp” triangles are ________________ triangles because…
I remember
this…
Some facts that I remember about these triangles…




ANOTHER way to label triangles involves focusing on one of the acute angles, and relating
the sides to that angle.
A
side
ADJACENT
to A
C
A
hypotenuse
side OPPOSITE toA
hypotenuse
side
OPPOSITE
to B
B
C
side ADJACENT to
B
B
Example:
The side opposite to B has a length of 10. This
side can also be described as adjacent to A.
The side opposite to A has length of 24. This side
can also be described as adjacent to B.
The hypotenuse is always the side opposite the
right angle that connects the two legs. The
hypotenuse in this triangle has a length of 26.
A
10
26
B
C
24
Summarize
The ratios that you looked at in the “ramp” situation have special names:

hypotenuse
opposite
The SINE (abbreviated sin) of angle is the ratio of the
side opposite that angle to the hypotenuse.
A
opposite
sin( A)=
hypotenuse
This shows us how big the side opposite angle A is compared to the hypotenuse.
If we know the measure of angle A, we can use the calculator to give us more information.
For example, if A is 25°, our calculator tells us
sin( 25)= 0.423
We can think about this number in a few different ways:
Since 0.423 =
423
1000
42.3
=
100
4.23
=
10
,
hypotenuse
we can see that if our opposite side was 423 units, our hypotenuse
opposite
would be 1000 units; or if our hypotenuse was 100 units, our
opposite side would be 42.3 units, and so on.
Since 0.423 = 42.3%, we can see that no matter the length of our
hypotenuse, our opposite side will be 42.3% of that length.
sin(65)=0.906
Example:
hypotenuse
This means the opposite side (to the 65°
angle) is 0.906 or 90.6% of the
hypotenuse.
65°
adjacent
20
90.6% of 20 = 0.906(20) = 18.12
opposite
So, the opposite side has a length of
18.12.
Summarize

The COSINE (abbreviated cos) of angle is the ratio of the side
hypotenuse
adjacent that angle to the hypotenuse.
This shows us how big the side adjacent angle A is
compared to the hypotenuse.
A
adjacent
If we know the measure of angle A, we can use the calculator to give us more
information. For example, if A is 25°, our calculator tells us
cos(25)= 0.906
adjacent
We can think about this number in a few different ways, as with sine.
For example, since 0.906 = 90.6%, we can see that no matter the length of
our hypotenuse, our adjacent side will be 90.6% of that length.
hypotenuse

The TANGENT (abbreviated tan) of an angle is the ratio of the side opposite an
angle to the side adjacent to that angle.
hypotenuse
This shows us how big the side opposite angle A is compared
to the side adjacent to angle A.
opposite
A
adjacent
If we know the measure of angle A, we can use the calculator
to give us more information. For example, if A is 25°, our calculator tells us
tan( 25)= 0.466
We can think about this number in a few different ways, as with sine and cosine.
For example, since 0.466 = 46.6%, we can see that no matter the length of
opposite
our adjacent side, our opposite side will be 46.6% of that length.
S
Sin =
OH
Opposite
Hypotenuse
C
Cos =
AH
Adjacent
Hypotenuse
T
Tan =
OA
Opposite
Adjacent
adjacent
Practice
1. Using XYZ, find the following ratios. Give your answers as fractions, decimals, AND percents.
a. sin X =
Z
5
Y
b. cos X =
c. tan X =
12
13
d. sin Y =
e. cos Y =
X
f. tan Y =
g. sin Z =
h. Why do you think we can’t find cos Z or tan Z ?
2. Write a statement for what sin X tells about the relationship between angle X, side ZY, and side
YX.
3. Use you calculator to find the values below. Write your answers as both decimals and percents.
Then write a statement telling what this value tells about the relationship between the angle and
certain sides in the triangle.
a. sin 30°=
c. sin 23°=
b. tan 45°=
d. sin 71°=
Summarize

Since the trigonometric ratios relate angles to sides (in right triangles), they can often be
used to find missing sides even when the Pythagorean Theorem cannot (that is, even if we are
given only one other side, as long as we are also given one other angle besides the 90°).
A
Example:
1
st
c
b
Find the missing sides and angle in the right triangle ABC.
42°
We want to identify what parts of the triangle we have, and
how they relate to one another:
We have our 90° angle, another angle of 42°, and a leg of 17 that is
NOT opposite our angle of 42° (in other words, we have a side
adjacent to our angle).
2nd Since we have an angle and a side Adjacent to that angle, we
can use either Cosine or Tangent (because both of those ratios
involve the Adjacent side).
adj
We can choose which side to find first!
cos(B)=
hyp
We’ll choose to find the
17
Hypotenuse first, so we’ll use Cosine:
cos(42°)=
c
cos(42°) 17
=
1
c
ccos(42°)=1 17
ccos(42°)
17
=
cos(42°)
cos(42°)
17
c=
cos(42°)
c = 22.88 units
C
17
A
c
b
C
42°
B
17
side ADJACENT to our angle
opp
tan(B)=
3rd
Then we need to find the last side. Since the remaining side is
Opposite our given angle, and we were given the Adjacent side, we’ll use
Tangent:
(OR we could use the Pythagorean Theorem now!)
4th
adj
b
tan( 42°)=
17
tan( 42°)
b
=
1
17
17tan( 42°)=1 b
17cos(42°)= b
15.31 = b
Finally, we need to find the remaining angle, we would do 180°-90°-42° = 48°.
B
Summarize

Since the trigonometric ratios relate angles to sides (in right triangles), they can also be
used to find angles, something the Pythagorean Theorem definitely cannot help us to do.
A
Example:
Find the missing side and angles in right triangle ABC.
1st
We want to identify what parts of
the triangle we have, and how they relate
another: Here, we already have two sides,
we can use the Pythagorean Theorem to
the missing side, c:
c
17
C
a2 + b 2 = c 2
172 + 24 2 = c2
289 + 576 = c 2
865 = c 2
to one
so
find
865 = c2
29.41  c
2nd Now we need to find one of the acute angles.
A
We can choose which angle to find first!
17
adjacent to A
We’ll choose to find angle A, so we need to relate
our given sides to angle A. In this case, we have the
Opposite and the Adjacent sides, so this tells us to use
Tangent:
C
3
We use tan-1, which we
tan( A)=
B
24
opposite to A
adj
24
17
tan -1 [tan( A)] = tan -1 [
read “tangent inverse” :
tan-1, cos-1, and sin-1 “undo” tan, cos, and sin.
These are used to FIND ANGLES!
4th
c
opp
tan(A)=
But how do we “move” the tan to the other
side of the equation to get A by itself?
rd
B
24
A = tan -1 [
A = 54.7°
We need to find the third angle, so we’ll do 180°-90°-54.7°= 35.3°.
24
17
24
17
]
]
Practice
For #1-4, SOLVE each of the triangles below. In other words, find the missing sides and angles.
SHOW YOUR WORK, and clearly label your final answers.
1.
2.
A
c
17
f
E
d
D
21
57°
37°
C
3.
B
a
G
14
F
4.
J
I
k
7
13
g
K
H
15
L
Apply
5. Re-consider the “ramp” problem from earlier. Recall that the top of the stairs was 2 feet off the
ground, the ground-to-ramp angle needed to be 4.76°.
a. Label the diagram below.
Q
R
P
b. Find the horizontal distance from the base of the stairs to the start of the ramp, to the
nearest tenth of a foot.
c. Find the length of the ramp, to the nearest tenth of a foot.
d. Another contractor wants to build a ramp that will go up a much higher staircase, the
top of which is 6 feet vertically above the ground. However, he only has 65 feet of ground
space (horizontal distance) available. Is this contractor legally allowed to build his ramp?
Source: http://www.hpw.gov.yk.ca/pm/883.html
Apply
6. While vacationing in Italy, you find yourself relaxing on the Piazza del Duomo in Pisa, lounging
on a blanket, looking up at the famous Leaning Tower.
a. You estimate (by pacing) that you are 35 ½ yards from the base of the tower, and that
your angle of elevation (from your eyes up to highest part of the top of the tower) is about
60°. Find the approximate height above ground of the highest part of the tower.
b. You estimate that you are about 52 yards from the base of the
tower on the other side, and that your angle of elevation (from
your eyes up to the lowest part of the top of the tower) is about
50°. Find the approximate height above ground of the lowest
part of the tower.
(a.)
60°
50°
35.5 yds
16.5 yds
c. When you stand directly beneath the lowest part
of the top of the tower, you realize that you are
actually about 3 ½ yards away from the base of the
tower, and you estimate that you can see all but the
top 10 yds of the tower. What angle is the tower
making with the ground?
(b.) – 10 yds.
3.5 yds
90° angle
between
you and the
ground
(b.)