So far we have used trigonometry to find unknown side lengths and angles in right-angled triangles.
Now, imagine we had to work out the height to the peak of a mountain, or the height of an airplane in the sky. Trigonometry can help with a lot of these problems. In fact, there are many examples of professions that use trigonometry:
- Surveyors measure an angle and a length, then use trigonometry to get another length that cannot be measured directly, such as a distance across a lake.
- Astronomers measure the lengths of shadows of mountains on the moon and knowing the sun angle can compute the heights of the mountains. They also use it when finding the distance between celestial bodies
- Architects use trigonometry to calculate structural load, roof slopes, ground surfaces and many other aspects, including sun shading and light angles
- Ship captains use trigonometry in navigation to find the distance of the shore from a point in the sea.
- Oceanographers use trigonometry when calculating the height of tides in oceans
In real life contexts, problems involving trigonometry usually involve the following steps in order to arrive at a solution:
- Draw a diagram of the situation
- Identify any right-angled triangles
- Determine any given information
- Work out the trigonometric ratio to use
- Solve the problem
Practice questions
Question 1
If $d$d is the distance between the base of the wall and the base of the ladder, find $d$d to two decimal places.
Question 2
Jack is standing at the tip of a tree's shadow and knows that the angle from the ground to the top of the tree is $34^\circ$34°.
If Jack is standing $29$29 metres away from the base of the tree, what is the value of $h$h, the height of the tree to the nearest two decimal places?
Angles of elevation and depression
Trigonometry is commonly used in problems involving special angles called angles of elevation and depression.
Angle of Elevation
An angle of elevation is the angle created when an observer is looking at an object which is above the horizontal. The angle between the horizontal and the observer's line of sight is called the angle of elevation.
Angle of Depression
An angle of Depression is the angle created when an observer is looking at an object which is below the horizontal. The angle between the horizontal and the observer's line of sight is called the angle of depression.
Angles of elevation or depression always have a horizontal line in them.
Angles of elevation are measured from the horizontal line upwards.
Angles of depression are measured from the horizontal line downwards.
Worked example
example 1
Jasper measured the distance from a point to the base of a tree and the angle of elevation from the same point to the top of the tree. Calculate the height of the tree.
Think: Drawing a diagram and marking on any given information is a useful strategy. We can then identify any right-angled triangles and use trigonometry to solve for the unknown height. The angle of elevation goes from the horizontal line on the ground in an upwards direction.
Do: We can see that we have an angle, the adjacent side and want to calculate the opposite side. Hence, we will use the tangent ratio.
| $\tan38^\circ$tan38° | $=$= | $\frac{\text{Height of tree }}{4.2}$Height of tree 4.2 m |
| $\text{height of tree }$height of tree | $=$= | $4.2\times\tan38^\circ$4.2×tan38° |
| $\text{height of tree }$height of tree | $=$= | $3.28$3.28 (to $2$2 d.p.) |
So the height of this tree is $3.28$3.28 m.
Practice questions
Question 3
A man standing at point $C$C, is looking at the top of a tree at point $A$A. Identify the angle of elevation in the figure given.
A right triangle is shown with vertices labeled A, B and C. Side AB is the vertical leg, side BC is the horizontal leg, and side AC is the hypotenuse. The right angle is at vertex B, as indicated by a small square. There are two arcs indicating angles: one at vertex C, labeled with $\alpha$α, and another at vertex A, labeled with $\theta$θ. Above vertex C, a vertical dotted line extends from vertex C to a point labeled D. The angle between this dotted line CD and the hypotenuse AC is labeled with $\sigma$σ.
$\alpha$α
A$\theta$θ
B$\sigma$σ
C
Question 4
At a certain time of the day a light post, $6$6 m tall, has a shadow of $5.8$5.8 m. If the angle of elevation of the sun at that time is $\theta$θ°, find $\theta$θ to two decimal places.
Question 5
From the top of a rocky ledge $188$188 m high, the angle of depression to a boat is $13^\circ$13°. If the boat is $d$d m from the foot of the cliff find $d$d correct to two decimal places.
Question 6
A ship is $27$27m away from the bottom of a cliff. A lighthouse is located at the top of the cliff. The ship's distance is $34$34m from the bottom of the lighthouse and $37$37m from the top of the lighthouse.
Find the distance from the bottom of the cliff to the top of the lighthouse, $y$y, correct to two decimal places.
Find the distance from the bottom of the cliff to the bottom of the lighthouse, $x$x, correct to two decimal places.
Hence find the height of the lighthouse to the nearest tenth of a metre.