Right Triangles and Trigonometry
Similar Triangles are characterized by congruent corresponding angles
and proportionate corresponding sides.
There are several distinct characteristics of Right Triangles that we will
look at in Chapter 9 that make them very useful in designing structures
and analyzing the world around us.
C
Theorem 9.1
If the altitude is drawn from the right angle to
the hypotenuse of a right triangle, then the
two triangles formed are similar to the original
triangle, and to each other
/\ CBD ~ /\ ABC, /\ ACD ~ /\ ABC, and
/\CBD ~ /\ACD
A
D
B
Finding the Height of a Roof
Y
The roof has a cross section that forms a
right triangle. How can we use that
information to determine the actual height
of the roof?
5.5
Z
h
3.1
W
1. Create a series of “similar” right triangles
Z
Z
Y
6.3
5.5
5.5
3.1
X
h
W
~
Y
~
h
W
X
3.1
Y
|-------------------------6.3---------------------------|
2. Then use the proportional component of similarity to determine the value of h
YW – XY
ZY XZ
_h_ -- 3.1
5.5 6.3
6.3h = 5.5(3.1) = 2.7
X
Using Geometric Mean to Solve Problems
C
Theorem 9.2
In a right triangle, the altitude from the right angle to the
hypotenuse divides the hypotenuse into two segments.
The length of the altitude is the GEOMETRIC MEAN of
the lengths of the two segments.
A
D
BD – CD
CD AD
Theorem 6.3
In a right triangle, the altitude from the right angle to the
hypotenuse divides the hypotenuse into two segments.
AB – CB
CB DB
The length of each leg of the right triangle is the
geometric mean of the lengths of the hypotenuse and
the segment of the hypotenuse that is adjacent to the
leg.
AB – AC
AC AD
B
Using Geometric Mean
2
x
y
6
6–x
x 3
18 = x2
X = \/18
5
3
5+2–y
y
2
7–y
Y 2
Y2 = 14
X = 3\/2
Y = \/14
Trigonometric Ratios
Triangle ABC is a right triangle. The sine (sin), cosine (cos) and tangent (tan)
are defined as follows:
sin A = side opposite <A = a
hypotenuse
c
B
Hypotenuse
c
cos A = side adjacent <A = b
hypotenuse
c
tan A = side opposite <A = a
side adjacent <A b
a
side opposite
<A
A
b
side adjacent <A
C
Finding Trig Ratios
Based upon the SSS Similarity
Theorem the two triangles are similar.
What impact do you think this will have
on the Trigonometric ratios?
B
B
17
8
C
4
15
A
C
8.5
7.5
A
Large Triangle
Small Triangle
sin A = side opposite <A = a
hypotenuse
c
8/17 = 0.4706
4/8.5 = 0.4706
cos A = side adjacent <A = b
hypotenuse
c
15/17 = 0.8824
15/17 = 0.8824
tan A = side opposite <A = a
side adjacent <A b
8/15 = 0.5333
4/7.5 = 0.533
Trig Ratios for 45o and 30o Angles
In a 45 – 45 – 90 Triangle, the Sin, Cos, and Tan
will always fall into a standard proportion to
one another based upon the standard
relationship between the sides (9.4).
1
\/ 2
45o
1
sin 45o = 1/ \/2 = \/2 / 2 = 0.7071
cos 45o = 1/ \/2 = \/2 / 2 = 0.7071
tan 450 = 1/1 = 1
In a 30 - 60 – 90 Triangle, the Sin, Cos, and
Tan will always fall into a standard proportion
to one another based upon the standard
relationship between the sides (9.4).
sin 30o = 1 / 2 = 0.5
1
2
30o
\/3
sin A = side opposite <A = a
hypotenuse
c
cos 30o = \/3 / 2 = 0.8660
tan
30o
= 1 / \/3 = \/3 / 3 = 0.574
cos A = side adjacent <A = b
hypotenuse
c
tan A = side opposite <A = a
side adjacent <A b
Solving for a Right Triangle
Solving for a Right Triangle means determining the measures of every
side and every angle
- We can do this if we know:
* The measure of 2 Sides, or
* The measure of 1 Side and 1 Angle
Solving knowing 1 Side & 1 Angle
Solving knowing 2 sides:
-If we know the measure of two
sides, we can calculate the Sin, Cos,
or Tan (depending on givens)
-Once we know that measure, we
can take the inverse of the measure
to determine the measure of the
angle
- Triangle ABC has leg
measures of 2 and 3
-Taking 2/3 we determine that
the TAN of angle A = .6
-The inverse of the Tan of .6 =
33.7
-Therefore the measure of
angle A = 33.7
-Therefore Triangle ABC has
side measures of 2, 3, and
Root 13; and angle measures
of 90, 33.7, and 56.3
B
2
C
3
Y
h
13
25o
Z
g
-If we know the measure of one angle
and one side, we can calculate the
length of the missing sides.
-Given we know the measure of the
acute angle, we can determine the
A missing angle by subtracting it from 90
-Triangle XYZ has one acute
angle that measures 25o and
a hypotenuse that measures
13
-Taking the Sin of X, we
determine the opposite side
to have a measure of 5.5
-Taking the Cos of X we
determine the adjacent side
to be 11.8.
X
-Therefore the triangle has side
measures of 5.5, 11.8, and
13; and angle measures of
25, 65, and 90
Trig Application – Glide Angles and Glide Ratios
Aerospace Design and landing
takes into account two
concepts based upon Basic Trig
Ratios
-Glide Angle:
The angle of approach taken
by an aircraft as it enters its
landing pattern
Using this angle, pilots and
ATC can safely guide a plan on
a steady descent to it’s landing
-Glide Ratio:
Mathematical calculation
based on the design of the
aircraft that under specific
conditions (primarily
determined by speed and
altitude)
Determines the distance a
plane will travel under “glide”
conditions
When an aircraft (Space Shuttle) is at an altitude of 15.7
Miles it is 59 miles away from the runway. What is the
glide angle of the approach? TAN x = 15.7 / 59 = 14.9o
When the shuttle is 5 miles away, it has increased it’s glide
angle to 19o. What is it’s altitude? TAN 19o = h/5 = 1.7 Miles