16.02 Angles of elevation and depression
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.
A vertical rectangle outlined with dashed lines has its vertices marked with solid dots and are labeled $A$A, $B$B, $C$C, and $D$D. A helicopter is positioned in vertex $B$B at the top left of the rectangle. A diagonal extends from vertex $B$B to vertex $D$D and forms two right-angled triangles. The length of diagonal $BD$BD is labeled $19$19 units. The $\angle CBD$∠CBD in the upper right-angled triangle is marked with a blue-outlined double arc and represents an angle of depression. The $\angle CBD$∠CBD is labeled as $x$x. The $\angle ABD$∠ABD in the lower right-angled triangle is marked with blue-outlined single arc. The $\angle ABD$∠ABD is labeled as $y$y. The top horizontal side $BC$BC is adjacent to $\angle CBD$∠CBD. The right vertical side $CD$CD is opposite $\angle CBD$∠CBD. The left vertical side $AB$AB is adjacent $\angle ABD$∠ABD. The bottom horizontal side $AD$AD is opposite $\angle ABD$∠ABD. The bottom horizontal side $AD$AD is labeled $9$9 units. The right angle at vertex $A$A of the lower right-angled triangle is marked with a small blue-shaded square. The right angle at vertex $C$C of the upper right-angled triangle is also marked with a small blue-shaded square.
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.
A right-angled triangle formed by a tree, the ground, and the line of sight from a cat looking up. On the left side, the tree is labeled with a vertical height of $1.36$1.36 m. The right angle between the height of the tree and the ground is marked with a small light orange square. To the right, a cat is positioned on the ground, looking up at the top of the tree. The angle of elevation between the ground and the cat’s line of sight is labeled $29^\circ$29° and is marked with a purple-shaded arc. The horizontal distance from the cat to the base of the tree is labeled $d$d.
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.
Write each line of working as an equation.
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°.
Two fighter jets are flying at the same height, positioned above an airport. The position of the first jet, which is farther from the airport, is marked with a solid dot and is labeled $A$A. The position of the second jet, which is closer from the airport, is marked with a solid dot and is labeled $B$B. The airport is represented with a solid dot. The position directly above the airport and is at the same height of the fighter jets is marked with a solid dot and is labeled $C$C. The vertical distance between point $C$C and airport is labeled as $4000$4000 m. From the fighter jets, two diagonal lines extend downward to the airport, representing lines of sight. The angle of depression from the first fighter jet is marked with a single blue-shaded arc and is labeled as $13^\circ$13°. The angle of depression from the second fighter jet is marked with a double purple-shaded arcs and is labeled as $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.