Question Video: Using Right-Angled Triangle Trigonometry to Solve Word Problems Involving Angles of Elevation and Depression

A building is 8 meters tall. The angle of elevation from the top of the building to the top of a tree is 44° and the angle of depression from the top of the building to the base of the tree is 58°. Find the distance between the base of the building and the base of the tree giving the answer to two decimal places.

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Video Transcript

A building is eight meters tall. The angle of elevation from the top of the building to the top of the tree is 44 degrees and the angle of depression from the top of the building to the base of the tree is 58 degrees. Find the distance between the base of the building and the base of the tree giving the answer to two decimal places.

So let’s begin by sketching this problem. First, we have a building which is eight meters tall. We’re then told about an angle of elevation from the top of this building to the top of a tree. Remember, an angle of elevation is measured from the horizontal up towards something. So this tree is taller than the building. So we add the tree and the angle of elevation of 44 degrees. Next, we’re told that the angle of depression from the top of the building to the base of the tree is 58 degrees. We need to be careful when labeling this angle. Remember that an angle of depression is measured from the horizontal down towards something, so the angle of 58 degrees is this angle here.

What we’re looking to calculate is the distance between the base of the building and the base of the tree, which is this length here. Now, looking carefully at our diagram, we can see that we have a right triangle. And the length we’re looking to calculate, which we’ll call 𝑦 meters, is one of the sides. We also know another of the sides. It’s the height of the building, eight meters. We can also work out one of the angles inside our triangle. If we subtract the angle of depression of 58 degrees from a right angle of 90 degrees, we can find the top angle in our triangle. It’s 32 degrees. So we now have a right triangle in which we know one length and one other angle, and we wish to calculate a second length.

Labeling the sides of the triangle in relation to the 32-degree angle, we know the adjacent and we want to calculate the opposite. So we’re going to use the tan ratio. Recall, the tan ratio is opposite over adjacent, so we have tan of 32 degrees is equal to 𝑦 over eight. Multiplying both sides of this equation by eight, we have that 𝑦 is equal to eight tan 32 degrees. Evaluating this on a calculator, which must be in degree mode, we have 4.9989.

We’re asked to give our answer to two decimal places. So starting with the eight in the third decimal place and rounding up, we have 5.00 meters. So we’ve completed the problem. The distance between the base of the building and the base of the tree to two decimal places is 5.00 meters.

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