Angles of elevation and depression

Angles of elevation and depression are angles that are given from above or below a horizontal perspective. For example the angle that is made when you look up at something is an angle of elevation. When you look down the angle that is made from your line of sight is called an angle of depression. It is important to note both of these angles are measured from a horizontal perspective.

Use this page to revise the following concepts within Angles of elevation and depression:

Angles of elevation and depression are alternate interior angles

Alternate interior angles are pairs of  equal angles formed on opposite sides of a transversal, but inside two parallel lines. Angles of elevation and depression are alternate interior angles. That is, they are a pair of equal angles formed on opposite sides of the transverse formed by the line of sight.  This can be illustrated as a Z-shaped figure.

There are two parallel horizontal lines AB (lower) and DC (upper). Two right angles are formed at points C and B, between DC and AB and the perpendicular dashed line CB. A transversal from A to C forms the hypotenuse of a right angled triangle ABC. The internal angle at points A and C are labelled theta, each formed between the transversal and the horizontal. The complementary alternate angle to theta formed from a line perpendicular to each horizontal is labelled alpha. This demonstrates that theta=90-alpha.

From the figure, we can see that a right angle is formed at the intersection of the line of sight and and a perpendicular segment \(C\). This right angle is made up of two complementary angles \(\theta\) and \(\alpha\).

The angles ,\(\theta\), are alternate interior angles and so are equal. Therefore, the angle of elevation is equal to the angle of depression.

Applications of angles of elevation and depression

Worked Example 1: Using the angle of elevation to find unknown sides

A park ranger measured the top of a tree to be at an angle of \(30^\circ\) when standing \(40\text{ m}\) from the base of the tree. To the nearest metre, what is the height of the tree?

1. Solve using trigonometry: first label triangle with Opp, Hyp, Adj given the location of angle in the right-angled triangle.

Observation of a tree. Observer is standing 40 metres from the base of the tree, and the angle formed with the top of the tree is 30°. The horizontal line has been labelled Adj, and a perpendicular line from the base to the top of the tree has been lablled Opp.

2. Identify the appropriate trigonometric ratio

\(\tan(\theta)=\frac{Opp}{Adj}\)

3. Substitute the values and rearrange the ratio to make the unknown side the subject.

\[\begin{align} Adj\times \tan(\theta)&=0pp \\ 40\times \tan(30^{\circ})&\approx23\text{ m} \end{align}\]

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Worked Example 2: Using the angle of depression to find unknown sides

From the top of an \(80\text{ m}\) cliff a spotter notices a shipwreck. The spotter incorrectly measures the angle of depression to be 25°. First find the angle of depression, then find the distance from the base of the cliff to the shipwreck to the nearest metre.

Observation of a boat from a raised cliff. Height of the cliff is 80m, and angle formed from the top of the cliff to the boat is 25°

1. Find angle of depression:

Angle of depression is measured off horizontal and is therefore \(65^\circ (90^\circ-25^\circ = 65^\circ). This is also the angle of elevation.

Observation of a boat from a raised cliff. Height of the cliff is 80m, and angle formed from the top of the cliff to the boat is 25°. The complementary angle from between the line of sight and a horizontal line extending from the clifftop is 65°, there the angle formed from the base of the cliff to the boat is also 65°Observation of a boat from a raised cliff. Height of the cliff is 80m, and angle formed from the top of the cliff to the boat is 25°. The complementary angle from between the line of sight and a horizontal line extending from the clifftop is 65°, there the angle formed from the base of the cliff to the boat is also 65°

2. Solve using trigonometry; first label triangle with Opp, Hyp, Adj given the location of angle in the right-angled triangle.

3. Identify the trigonometric ratio

\(\tan(\theta)=\frac{Opp}{Adj}\)

Substitute and rearrange ratio to make unknown side the subject. i.e

\[\begin{align} Adj &=\frac{Opp}{\tan(\theta)}\\ x&=\frac{80}{\tan(65^{\circ})}\\ &\approx37\text{ m} \end{align}\]