Trigonometry Tutorial
* Right Triangles
* Tangents
* Small Angle Approximation
* Triangulation
* Parallax
Triangles
θ
Triangles
Opposite
W
θ
Triangles
θ
d
Adjacent
Triangles
w
θ
d
Tan θ = Opposite/Adjacent = w/d
Similar Triangles
w
h
θ
L
d
The angle θ is the same for both triangles shown.
Therefore, Tan θ = w/d = h / L
Small Angle Approximation
If the angle θ is small, then
Tan θ ≈ θ
If the Small Angle Approximation is used,
θ will be in radians
Triangles - Example
1m
θ
4 cm
What is the angle θ for the triangle shown?
Tan θ = 1 cm/4 cm = .25
θ = Tan-1 .25 = 14o
Triangles
w
25o
6m
What is the length of the opposite side of the triangle shown?
Tan 25o = w /6 m
w = (6 m) Tan 25o = 2.79 m
Triangles
w
1 cm
3 cm
8 cm
What is the length of the opposite side of the larger triangle shown?
Tan θ = 1 cm/3 cm = w / 8 cm
w = (1/3) 8 cm = 2.67 cm
Measuring Angular Diameter of
Objects
θ
The angle θ is called the
“Angular Diameter”
or the
“Angular Size”
of the object.
Measuring Angular Diameter
w
θ
d
Measuring Angular Diameter
w
θ
d
Tan θ = Opposite/Adjacent = W/d
Measuring Angular Diameter
w
θ
d
What if you can’t measure w and d directly???
Measuring Angular Diameter
w
h
d
θ
L
What if you can’t measure w and d directly???
Use a measurable h and L
Measuring Angular Diameter
w
h
d
θ
L
What if you can’t measure w and d directly???
Use a measurable h and L
Then Tan θ = h/L = W/d
Measuring Angular Diameter
w
h
d
L
A tourist wishes to determine the height of the pagoda shown. It is known that tourist is
standing 300 m from the pagoda. The tourist uses a similar triangle to make an estimate
of the height of the pagoda, and determines that h = 1 cm and L = 15 cm. Based on this
information, what is the height of the pagoda?
h/L = w/d → 1 cm/15 cm = w/300 m → w = (1/15) 300 m = 20 m
Measuring Angular Diameter
w
h
θ
d
L
What is the angular diameter of the pagoda in the previous example?
Tan θ = 20 m/300 m = 0.0667
θ = Tan-1 0.067 = 3.81o
Measuring Angular Diameter
w
h
d
θ
L
How accurate is the Small Angle Approximation if it is used to determine the angular
diameter of the pagoda in the previous example?
Tan θ ≈ θ = 20 m/300 m = 0.0667 radians
From the previous calculation θ = 3.81o = 0.0665 radians
Although 4.76o might not be considered to be a “small angle,” 3.81o is small enough for the
Small Angle Approximation to be quite accurate.
Measuring Angular Diameter
Convenient measure used by people “in the field”
Use your hand as a standard.
For the average person:
At arms length,
* the angel between the tip of the pointer finger and the first knuckle
is about 2 degrees.
* the angle between the first knuckle and the second knuckle is
about 3 degrees.
* the angle between the second knuckle and the third knuckle is
about 4 degrees.
Measuring Angular Diameter
w
h
d
θ
L
The tourist in the pagoda example uses the “knuckle rule” (last slide) to estimate the
angular diameter of the pagoda. How would the pagoda compare with the knuckles?
The pagoda would extend almost from the second to the third
knuckle.
Parallax
Parallax (angle)
D
W
W/2
Parallax
Parallax (angle)
D
W/2
Parallax
Tan θ/2 = (W/2)/D
Parallax (θ)
D
W/2
Parallax
An astronomy student would like to determine the distance from the
earth to the moon using parallax arguments. By contacting
observatories on opposite sides of the earth, the student determines
that the parallax for the moon is 1.9o . Knowing that the radius of the
earth is 6378 km, what is the distance from the earth to the moon.
Tan 1.9/2 = (W/2)/D
D = 6378 km /(Tan 0.95o ) = 6378 km/0.00166 = 384630 km