9.07 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

The angle of elevation from a cat to the top of a tree is 35\degree. The distance between the tree and the observer is d metres and the tree is known to be 1.24 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 35 \degree, the length of opposite side is 1.24, 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 {35\degree}	\displaystyle =	\displaystyle \frac{1.24}{d}	Substitute the values
\displaystyle d \times \tan {35\degree}	\displaystyle =	\displaystyle 1.24	Multiply both sides by d
\displaystyle d	\displaystyle =	\displaystyle \frac{1.24}{\tan {35\degree}}	Divide both sides by \tan {35\degree}
	\displaystyle \approx	\displaystyle 1.77 \text{ m}	Evaluate using a calculator

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Usually when we find the value of an angle, we will get some decimal value in degrees and leave it at that. For more accuracy, we can measure our angles in degrees, minutes and seconds.

However, if you look at your calculator, you might see a button with \, \degree \, ' \, '' \, on it.

This button can be used to convert between decimal degrees and degrees, minutes and seconds on your calculator. It also shows us the notation for writing degrees, minutes and seconds.

Looking at the button, we can see three symbols which are used for degrees, minutes and seconds respectively.

For example: we can write 37 degrees, 12 minutes and 49 seconds as: 37\degree 12\rq 49\rq\rq

To convert a decimal value of degrees into minutes and seconds, we need to know that:

• 1 \text{ degree} = 60 \text{ minutes}

• 1 \text{ minute} = 60 \text{ seconds}

These conversion factors also tell us that the number of minutes or seconds should never be greater than 60 in a fully simplified expression.

Examples

Example 2

Find the angle of depression from point B to point D. Give your answer in degrees and minutes, rounding to the nearest minute.

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=7 and hypotenuse BD=19. 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{7}{19}	Subsitute the values
\displaystyle x	\displaystyle =	\displaystyle \cos^{-1}\left(\dfrac{7}{19}\right)	Apply inverse cosine on both sides
	\displaystyle =	\displaystyle 68.382\degree	Evaluate using a calculator
	\displaystyle =	\displaystyle 68\degree + 0.382 \times 60 \text{ minutes}	Convert the decimal to minutes
	\displaystyle \approx	\displaystyle 68\degree 23\rq	Evaluate the multiplication

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Example 3

At a certain time of the day a light post, 5 m tall, has a shadow of 9.3 m. Calculate the value of \theta, the angle of elevation of the sun at that time. Give your answer in degrees, minutes and seconds to the nearest second.

Worked Solution

Create a strategy

Use the tangent ratio.

Apply the idea

With respect to the angle \theta, the length of opposite side is 5, and the length of adjacent side is 9.3, so we can use the tangent ratio.

\displaystyle \tan \theta	\displaystyle =	\displaystyle \frac{\text{Opposite }}{\text{Adjacent}}	Use the tangent ratio
\displaystyle \tan \theta	\displaystyle =	\displaystyle \frac{5}{9.3}	Substitute the values
\displaystyle \theta	\displaystyle =	\displaystyle \tan^{-1}\left(\dfrac{5}{9.3}\right)	Apply inverse tangent on both sides
	\displaystyle =	\displaystyle 28.264\degree	Evaluate using a calculator
	\displaystyle =	\displaystyle 28\degree + 0.264 \times 60 \text{ minutes}	Convert the decimal to minutes
	\displaystyle =	\displaystyle 28\degree 15.84\rq	Evaluate the multiplication
	\displaystyle =	\displaystyle 28\degree 15\rq + 0.84 \times 60 \text{ seconds}	Convert the decimal to seconds
	\displaystyle \approx	\displaystyle 28\degree 15\rq 50\rq\rq	Evaluate the multiplication

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