In order to measure the height of a tree, Dan calculated that its shadow is \(12\;[m]\) long and that the line joining the top of the tree to the tip of the shadow forms an angle of \(52°\). Find the height of the tree to the nearest metre.

Show Answer
We can begin by drawing a diagram to help illustrate the problem:

As we are taking into account the opposite and adjacent sides of the triangle, we can use the \(\tan\) trigonometric function to solve for \(x\):

\(\tan52 = \cfrac{x}{12}\)

\((12)(\tan52) = (12)(\cfrac{x}{12})\)

\(x = (12)(\tan52)\)

\(x = 15.36\;[m] = 15.4\;[m]\)
Therefore, we can determine that the height of the tree is \(15.4\;[m]\).

At the top of a hiking trail, there are 2 vertical posts. One is \(5\;[m]\) tall and the other is \(7\;[m]\) tall. The ground between the posts is level, and the bases of the posts are 4m apart. The posts are connected by 2 straight wires.

i. What angle does each wire make with the ground?

ii. What is the length of each wire?

Show Answer
i. As we have the opposite and adjacent side lengths for both triangles, we can determine the angle each wire makes with the ground using the \(\tan\) trigonometric function:

\(\text{Wire}_{1}: \tan\theta = \cfrac{5}{4}\)

\(\text{Wire}_{1}: \theta = \tan⁻¹(\cfrac{5}{4})\)

\(\text{Wire}_{1}: \theta = 51°\)

\(\text{Wire}_{2}: \tan\theta = \cfrac{7}{4}\)

\(\text{Wire}_{2}: \theta = \tan⁻¹(\cfrac{7}{4})\)

\(\text{Wire}_{2}: \theta = 60°\)
Therefore, we can determine that Wire's 1 and 2 make respective angles of \(51°\) and \(60°\) with the ground.

ii. We can determine the lengths of each wire using the Pythagorean Theorem:

\(\text{Wire}_{1}: c^2 = (5)^2 + (4)^2\)

\(\text{Wire}_{1}: c^2 = 25 + 16\)

\(\text{Wire}_{1}: \sqrt{c^2} = \sqrt{41}\)

\(\text{Wire}_{1}: c = 6.4\;[m]\)

\(\text{Wire}_{2}: c^2 = (7)^2 + (4)^2\)

\(\text{Wire}_{2}: c^2 = 49 + 16\)

\(\text{Wire}_{2}: \sqrt{c^2} = \sqrt{65}\)

\(\text{Wire}_{2}: c = 8.1\;[m]\)
Therefore, we can determine that the lengths of Wire 1 and Wire 2 are \(6.4\;[m]\) and \(8.1\;[m]\) respectively.

Aimee and Russell are facing each other on opposite sides of an \(8\;[m]\) telephone pole. From Aimee's point of view, the top of the telephone pole is at an angle of elevation of \(52°\). From Russell's point of view, the top of the telephone pole is at an angle of elevation of \(38°\). How far apart are Aimee and Russell?

Show Answer
First, we can sketch a diagram to help visualize the problem:

As we already have the opposite side length determined, and since we are trying to determine the adjacent side lengths on both sides, we can use the tan trigonometric function to determine the total distance:

\(\text{Aimee}: \tan52° = \cfrac{8}{x}\)

\(\text{Aimee}: (x)(\tan52°) = (\cfrac{8}{x})(x)\)

\(\text{Aimee}: \cfrac{\tan52°x}{\tan52°} = \cfrac{8}{\tan52°}\)

\(\text{Aimee}: x = 6.25\;[m]\)

\(\text{Russell}: \tan38° = \cfrac{8}{y}\)

\(\text{Russell}: (y)(\tan38°) = (y)(\cfrac{8}{y})\)

\(\text{Russell}: \cfrac{\tan38°y}{\tan38°} = \cfrac{8}{\tan38°}\)

\(\text{Russell}: y = 10.24\;[m]\)

\(\text{Total Distance} = 10.24\;[m] + 6.25\;[m]\)

\(\text{Total Distance} = 16.49\;[m]\)
Therefore, we can determine that the total distance between Aimee and Russell is \(16.49\;[m]\).

From the top of the building, the angle of elevation of the top of a nearby building is \(28°\) and the angle of depression of the bottom of a nearby building is \(48°\). The distance between the 2 buildings is \(50\;[m]\). What is the height of the taller building?

Show Answer
Since we have the adjacent side length determined, we need to determine what the individual opposite side lengths are for both triangles. To do this, we can use the tan trigonometric function:

\(T_{1}: \tan28 = \cfrac{x}{50}\)

\(T_{1}: (\tan28)(50) = (\cfrac{x}{50})(50)\)

\(T_{1}: x = 26.59\;[m]\)

\(T_{2}: \tan48 = \cfrac{y}{50}\)

\(T_{2}: (\tan48)(50) = (\cfrac{y}{50})(50)\)

\(T_{2}: y = 55.53\;[m]\)

\(\text{Total Height} = 26.59\;[m] + 55.53\;[m]\)

\(\text{Total Height} = 82.12\;[m]\)
Therefore, we can determine that the height of the taller building is \(82.12\;[m]\).