Download Right Triangle Trigonometry

Right Triangle Trigonometry
Sections 9.1 and 9.2
Objective:
To use the sine, cosine, and tangent ratios to
determine missing side lengths in a right triangle.
IN A RIGHT TRIANGLE….
There are ratios we can use to determine side
lengths. These ratios are constant, no matter what
the lengths for the sides of the triangle are. These
ratios are called trigonometric ratios.
Three of the trigonometric ratios are:
 Sine (sin)
 Cosine (cos)
 Tangent (tan)
TRIG RATIOS
adjacent
opposite
B
SIN = leg opposite of angle
hypotenuse
Hypotenuse
opposite
adjacent
C
, O 
COS= leg adjacent to angle,
hypotenuse
TAN= opposite leg,  O 
 
adjacent leg  A 
SOHCAHTOA
 
H
 A
 
H
Write the trig ratios for the following:
B
z
A
y
x
C
tan A =
tan B=
sin A =
sin B =
cos A =
cos B =
Let’s put numbers in…
Use the triangle to write each ratio.
sin G =
G
cos T=
12
R
20
16
tan G =
T
sin T=
cos G=
tan T=
Example
Use the triangle to write each ratio.
X
63
Z
60
87
Y
If given the angle measure, you can use a trig function to
find a missing side length of a right triangle.
Find x.
M
Which trig ratio relates the
given angle, and the 2 sides??
57°
25
Set up equation:
L
x
K
Examples: Find x.
1.
4 30°
x
2.
x
58°
18
Example:
To measure the height of a tree, Mrs. Shattuck walked 125 ft.
from the tree, and measured a 32˚ angle from the ground to the
top of the tree. Estimate the height of the tree.
Example:
A 20 ft wire supporting a flagpole forms a 35˚ angle with the
flagpole. To the nearest foot, how high is the flagpole?
Example:
You are at the playground, and they just put in an awesome new
slide. The slide is 25 ft long, and it creates a 57˚ angle with the
ground. How high off the ground is the top of the slide?
If you need to find an angle in a right triangle given the side lengths,
you use the inverse of the trig function: tan-1, sin-1, cos-1
tan-1 (.5) = x
“The angle,x, whose tangent is .5”
sin-1(.7314)=x
“The angle,x, whose sine is .7314”
cos-1(.5592)=x
“The angle,x, whose cosine is .5592”
Fill in the blanks….
1.
cos __________ ≈ .0175
2.
sin __________ ≈ .9659
3.
tan___________ ≈ .2309
In your calculator, enter
cos-1(.0175)
Find the indicated angle measures.
R
1.
mR 
41
How would you now find the
measure of angle T??
S
T
47
2.
41
A
17
mA 
Find the measure of angle A.
A
15
32
Example:
A right triangle has a leg 1.5 units long and a hypotenuse 4.0
units long. Find the measures of its acute angles to the nearest
degree.
Example:
You are 200 ft from the base of a 150 ft building. What is the
angle formed from the ground where you are standing to the
top of the building??
Angles of Elevation and Depression
 Angle of Elevation-- the
angle that an observer would
raise his or her line of sight
above a horizontal line in
order to see an object.
 Angle of Depression-- If
an observer were above and
needed to look down, the
angle of depression would be
the angle that the person
would need to lower his or
her line of sight.
*Why are the angles of elevation and depression between the same
two objects congruent?*
Describe each angle as it relates to the
situation
1
2
3
4
Using angles of elevation and
depression
 You see a rock climber on a cliff at a 32° angle of
elevation. The horizontal ground distance to the cliff is
1000ft. Find the line of sight distance to the climber.
A surveyor stands 200 ft from a building to
measure its height. The angle of elevation to
the top of the building is 35°. How tall is the
building?
An airplane pilot sights a life raft at a
26°angle of depression. The airplane’s
altitude is 3km. What is the airplane’s
surface distance from the raft?
Miss Long sits in a treehouse 20ft above
the ground. She spots a calculator on the
ground with a 15°angle of depression.
What is the sight distance between Miss
Long and the calculator?
The world’s tallest unsupported flagpole towers
282 feet above the ground in Surrey, British
Columbia. The shortest shadow cast by the
flagpole is 137ft long. What is the angle of
elevation of the sun when the shadow is cast?