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:
In real life contexts, problems involving trigonometry usually involve the following steps in order to arrive at a solution:
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.
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?
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.
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.
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.
$\alpha$α
$\theta$θ
$\sigma$σ
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.
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.
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.