5.06 Angles of elevation and depression

An angle of elevation is the angle from the lower object to the higher one, while an angle of depression is the angle from the higher object to the lower one. Both angles are measured with respect to the horizontal plane of the reference object.

The angle of elevation from point A to B is the angle between the horizontal line at A and the line connecting the two points.

The angle of depression from point B to A is the angle between the horizontal line at B and the line connecting the two points.

Notice that the angle of elevation between two points will always be equal to the angle of depression between those two points, since they are alternate angles on parallel lines (since all horizontal planes will be parallel).

Combining the angles of elevation or depression between two objects with trigonometry can help us to solve problems involving missing lengths or angles.

When given the angle of elevation or depression between two objects, we will always be able to model their relative position using a right-angled triangle. Using trigonometry, if we are given any side length of this triangle then we can solve for the other side lengths in the triangle.

Alternatively, there are three distances between two objects: horizontal distance, vertical distance and direct distance. These will represent the adjacent, opposite and hypotenuse sides respectively, and if any two are given then we can find the angle of elevation and depression.

Examples

Example 1

Find the angle of depression from point B to point D. Use x as the angle of depression and round your answer to the nearest degree.

Worked Solution

Create a strategy

Identify where the angle of depression is and apply the appropriate trigonometric ratio.

Apply the idea

The angle of depression is the angle \angle CBD, with adjacent side BC=8 and hypotenuse BD=17. We can let x be the unknown angle.

\displaystyle \cos x	\displaystyle =	\displaystyle \dfrac{\text{Adjacent}}{\text{Hypotenuse}}	Use the cosine ratio
\displaystyle \cos x	\displaystyle =	\displaystyle \dfrac{8}{17}	Subsitute the values
\displaystyle x	\displaystyle =	\displaystyle \cos^{-1}\left(\dfrac{8}{17}\right)	Apply inverse cosine on both sides
	\displaystyle \approx	\displaystyle 62\degree	Evaluate using a calculator

Watch question walkthrough

Example 2

The angle of elevation from an observer to the top of a tree is 32\degree. The distance between the tree and the observer is d metres and the tree is known to be 1.36 m high. Find the value of d to 2 decimal places.

Worked Solution

Create a strategy

Use the tangent ratio.

Apply the idea

With respect to the given angle 32 \degree, the length of opposite side is 1.36, and the length of adjacent side is d, so we can use the tangent ratio.

\displaystyle \tan \theta	\displaystyle =	\displaystyle \frac{\text{Opposite }}{\text{Adjacent}}	Use the tangent ratio
\displaystyle \tan {32\degree}	\displaystyle =	\displaystyle \frac{1.36}{d}	Substitute the values
\displaystyle d \times \tan {32\degree}	\displaystyle =	\displaystyle 1.36	Multiply both sides by d
\displaystyle d	\displaystyle =	\displaystyle \frac{1.36}{\tan {32\degree}}	Divide both sides by \tan {32\degree}
	\displaystyle \approx	\displaystyle 2.18 \text{ m}	Evaluate using a calculator

Watch question walkthrough