20.02 Angles of elevation and depression (Extended)
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.
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.
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.
Practice questions
Question 1
Find $x$x, the angle of depression from point $B$B to point $D$D.
Round your answer to two decimal places.
Question 2
The angle of elevation from an observer to the top of a tree is $29^\circ$29°. The distance between the tree and the observer is $d$d metres and the tree is known to be $1.36$1.36 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, $6$6 m tall, has a shadow of $9.7$9.7 m. If the angle of elevation of the sun at that time is $\theta$θ, find $\theta$θ to two decimal places.
Question 4
A fighter jet, flying at an altitude of $4000$4000 m is approaching a target. At a particular time the pilot measures the angle of depression to the target to be $13^\circ$13°. After a minute, the pilot measures the angle of depression again and finds it to be $16^\circ$16°.
Find the distance $AC$AC.
Round your answer to the nearest metre.
Find the distance $BC$BC.
Round your answer to the nearest metre.
Now find the distance covered by the jet in one minute.
Round your answer to the nearest metre.