Angles of elevation and depression

If a person stands and looks up at an object, the angle of elevation is the angle between the horizontal line of sight and the object.

Person with line of sight, object, angle of elevation and horizontal labelled

If a person stands and looks down at an object, the angle of depression is the angle between the horizontal line of sight and the object.

Person with object, angle of depression and horizontal labelled

Trigonometry can be used to solve problems that use an angle of elevation or depression.

Example

An architect wants to calculate the height of a building. He stands 50 m away from the base of the building and looks up at the top of the building. The angle of elevation from the architect to the top of the building is 70°. Calculate the height of the building. Give the answer to one decimal place.

Triangle at 70degrees 50m from a skyscraper

Label the sides of the triangle o, a and h.

Next choose the correct ratio from s^o_h~c^a_h~t^o_a.

In the triangle the length a is known and the length o must be calculated.

Use \tan{x} = \frac{o}{a}

\tan{70} = \frac{z}{50}

Make z the subject by multiplying both sides by 50.

z = 50 \times \tan{70}

z = 137.4~\text{m}

The building is 137.4 m tall.

Question

From the top of a 72 m high vertical cliff, a boat has an angle of depression of 32°. How far is the boat from the base of the cliff? Give the answer to 1 decimal place.

Triangle showing distance of yacht from coastline

In the triangle the length o is known and the length a must be calculated.

Use \tan{x} = \frac{o}{a}

\tan{32} = \frac{72}{y}

Rearrange the equation to make y the subject.

Multiply both sides by y.

y \times \tan{32} = 72

Divide both sides by \tan{32}.

y = \frac{72}{\tan{32}}

y = 115.2~\text{m}

The boat is 115.2 m from the base of the cliff.