Transcript angle of depression

Acc. Alg./Geom. A
UNIT QUESTION: What patterns
can I find in right triangles?
Standard: MM2G1, MM2G2
Today’s Question:
How do you use trig ratios to find all
the missing parts of a triangle?
Standard: MM2G2.a,b
Opp Leg
Sin 
Hyp
Adj Leg
Cos 
Hyp
Opp Leg
Tan 
Adj Leg
hypotenuse

adjacent
opposite
opposite
Using a Calculator
If you know the value of a specific trig
ratio for an unknown angle, you can
calculate the measure of the angle.
For example, if for
the triangle below
3
8
SinB  .375
SinB 
8
3
B
On most calculators written above the
Sin, Cos and Tan buttons are:
Sin 1 , Cos 1 , Tan 1
These are used to find the angle when you already know the value for the
ratio. On the calculator there will be a button, sometimes it reads “2nd”, that
will need to be pushed before you push the Sin, Cos or Tan button. This
button will allow you to do the function above the button.
Using a Calculator
3
SinB 
8
SinB  .375
On the calculator enter 0.375 the hit the
“2nd” button and then the “Sin” button.
Sin 1 0.375  22.02
The number 22.02 should be displayed.
This is the angle that has a Sin value of
0.375
Then, that means
8
3
B
On a graphics calculators you
will enter it just like it reads in
the equation.
Sin
1
 0.375
B  22.02
Then you can calculate the
angle value.
Problem-Solving Strategies
Scenario 1) You are given 2 sides of the triangle.
Find the other side and the two non-right angles.
B
G
c
13
OR
15
C
A
C
20
1A. Use the Pythagorean
theorem to find the 3rd side.
c 2  152  202
c  25
15
20
 15 
A  Tan 1  
 20 
A  36.9
B  53.1
TanA 
1B. Use an inverse trig function to
get an angle. Then use that angle
to calculate the 3rd angle. Sum of
the angles = 180º
k
K
12
k 2  122  132
k  25
12
13
 12 
K  Cos 1  
 13 
K  22.6
G  67.4
CosK 
Problem-Solving Strategies
Scenario 1) You are given 2 sides of the triangle.
Find the other side and the two non-right angles.
B
G
c
13
OR
15
C
k
A
C
20
2A. Use an inverse trig function to
get an angle. Then use the sum
of the angles = 180º to find the
3rd angle.
2B. Use a trig ratio using one of
the two angles to get the 3rd
side.
15
20
 15 
A  Tan 1  
 20 
A  36.9
B  53.1
TanA 
Cos36.9 
20
c
20
Cos36.9
c  25
c
K
12
12
13
 12 
K  Cos 1  
 13 
K  22.6
G  67.4
CosK 
Tan22.6 
a
20
a  20  Tan22.6
a5
Problem-Solving Strategies
B
Scenario 2) You are given an angle
and a side.
Find the other angle and the two other
sides.
1A. Use 2 different trig ratios from
the given angle to get each of
the other two sides.
1B. Use the sum of the
angles to get the 3rd angle.
26
51
a
C
a
26
0.6293  26  a
16.4  a
Cos51 
A
b
b
26
0.7771 26  b
20.2  b
Sin51 
A  180 (90 51)
A  39
Problem-Solving Strategies
Scenario 2) You are given an
angle and a side.
Find the other angle and the
two other sides.
2A. Use the sum of the
angles to get the 3rd angle.
2B. Use 2 different trig ratios from
the 3rd angle to get each of
the other two sides.
B
26
51
a
C
A
b
A  180 (90 51)
A  39
b
Sin39 
26
0.6293  26  b
16.4  b
a
Cos39 
26
0.7771 26  a
20.2  a
Problem-Solving Strategies
B
Scenario 3) You are given all 3
sides of the triangle.
Find the two non-right angles.
25
7
C
1. Use 2 different trig ratios to
get each of the angles.
A
24
24
CosA 
25
1  24 
A  Cos  
 25 
A  16.3
24
TanB 
7
1  24 
B  Tan  
7
B  73.7
Problem-Solving Strategies
B
Scenario 3) You are given all 3
sides of the triangle.
Find the two non-right angles.
25
7
C
2A. Use a trig ratio to get one
angle.
2B. Use the sum of angles to
get the 3rd angle
A
24
24
25
1  24 
A  Cos  
 25 
A  16.3
CosA 
B  180  (90  16.3)
B  73.7
Angle of Elevation/Depression
Sometimes when we use right triangles to model real-life situations, we
use the terms angle of elevation and angle of depression.
If you are standing on the ground and looking up at a hot air balloon, the
angle that you look up from ground level is called the angle of elevation.
If someone is in the hot air balloon and looks down to the ground to see
you, the angle that they have to lower their eyes, from looking straight
ahead, is called the angle of depression.
Balloon
Angle of
depression
Angle of
elevation
You
Angle of Elevation/Depression
If you look up 15º to see the balloon, then the person in the
balloon has to look down 15º to see you on the ground.
Angle of elevation = Angle of depression.
Balloon
Angle of depression = 15º
Angle of
elevation= 15º
You
Notice that in this situation, the one of the legs that forms the
right angle is also the height of the balloon.
Draw a Picture
When solving math problems, it can be very helpful to
draw a picture of the situation if none is given.
Here is an example.
Find the missing sides and angles for
Triangle FRY. Given that angle Y is
the right angle, f = 68, and y = 88.
F
88
r
The picture helps to visualize what
we know and what we want to find!
Y
R
68