Trigonometry Tutorial

* Right Triangles

* Tangents

* Small Angle Approximation

* Triangulation

* Parallax

Triangles

θ

TrianglesOpposite

W

θ

Triangles

Adjacent

d

θ

Triangles

w

Tan θ = Opposite/Adjacent = w/d

d

θ

Similar Triangles

w

d

h

L

The angle θ is the same for both triangles shown.

Therefore, Tan θ = w/d = h / L

θ

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

What is the angle θ for the triangle shown?

Tan θ = 1 cm/4 cm = .25

θ = Tan-1 .25 = 14o

1 m

4 cm

θ

Triangles

25o

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

6 m

w

Triangles

1 cm

8 cm

3 cm

w

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

The angle θ is called the “Angular Diameter”

or the “Angular Size”

of the object.

θ

Measuring Angular Diameter

d

w

θ

Measuring Angular Diameter

d

w

Tan θ = Opposite/Adjacent = W/d

θ

Measuring Angular Diameter

d

w

What if you can’t measure w and d directly???

θ

Measuring Angular Diameter

d

w

What if you can’t measure w and d directly???

Use a measurable h and L

h

L

θ

Measuring Angular Diameter

d

w

What if you can’t measure w and d directly???

Use a measurable h and L

Then Tan θ = h/L = W/d

h

L

θ

Measuring Angular Diameter

d

w

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

h

L

Measuring Angular Diameter

d

w

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

h

L

θ

Measuring Angular Diameter

d

w

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 radiansFrom 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.

h

L

θ

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

d

w

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.

h

L

θ

ParallaxParallax

Parallax (angle)

W/2

D

W

ParallaxParallax

Parallax (angle)

W/2

D

ParallaxParallax

Parallax (θ)

W/2

D

Tan θ/2 = (W/2)/D

ParallaxParallax

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