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8.07 Angles of elevation and depression

Lesson

One aspect of applying mathematics to the real world is coming up with names for all the different measurements so that we can keep track of what all our numbers mean.

Angles of elevation and depression are the angles between objects at different heights.

 

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.

Summary

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

Summary

The angle of depression from point $B$B to $A$A is the angle between the horizontal line at $B$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).

 

Using angles to solve problems

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.

 

Writing degrees, minutes and seconds

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 that looks like this .

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$37 degrees, $12$12 minutes and $49$49 seconds as:

$37^\circ12'49"$37°1249"

 

Converting to degrees, minutes and seconds

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

  • $1$1 degree $=$=$60$60 minutes
  • $1$1 minute $=$=$60$60 seconds

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

Worked example

Convert $24.759^\circ$24.759° into degrees, minutes and seconds.

Think: We already have $24$24 whole degrees, so we just need to convert $0.759^\circ$0.759° into minutes and seconds.

Do: We know that $1^\circ$1° is equal to $60$60 minutes, so:

$0.759$0.759 degrees $=$=$0.759\times60$0.759×60 minutes $=$=$45.54$45.54 minutes

This tells us that we have $45$45 whole minutes, with the remaining $0.54$0.54 minutes to be converted into seconds.

Since $1$1 minute is equal to $60$60 seconds:

$0.54$0.54 minutes $=$=$0.54\times60$0.54×60 seconds $=$=$32.4$32.4 seconds

Therefore, $24.759^\circ$24.759° is equal to $24$24 degrees, $45$45 minutes and $32.4$32.4 seconds. Using the degrees, minutes and seconds notation, we can write this as:

$24.759^\circ=24^\circ45'32.4"$24.759°=24°4532.4"

Practice questions

Question 1

Find the angle of depression from point $B$B to point $D$D.

Use $x$x as the angle of depression and give your answer in degrees and minutes to the nearest minute.

Question 2

The angle of elevation from an observer to the top of a tree is $35^\circ$35°. The distance between the tree and the observer is $d$d meters and the tree is known to be $1.24$1.24 m high. Find the value of $d$d to $2$2 decimal places.

Question 3

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

Question 4

A fighter jet, flying at an altitude of $3000$3000 m is approaching a target. At a particular time the pilot measures the angle of depression to the target to be $12^\circ$12°. After a minute, the pilot measures the angle of depression again and finds it to be $15^\circ$15°.

  1. Find the distance $AC$AC.

    Round your answer to the nearest metre.

  2. Find the distance $BC$BC.

    Round your answer to the nearest metre.

  3. Now find the distance covered by the jet in one minute.

    Round your answer to the nearest metre.

Outcomes

MA5.2-13MG

applies trigonometry to solve problems, including problems involving bearings

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