Question Video: Using the Law of Sines to Calculate an Unknown Length Involving Angles of Depression | Nagwa Question Video: Using the Law of Sines to Calculate an Unknown Length Involving Angles of Depression | Nagwa

# Question Video: Using the Law of Sines to Calculate an Unknown Length Involving Angles of Depression Mathematics • Second Year of Secondary School

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The angle of depression of a car parked on the ground from the top of a hill is 48°. A view point is 14 meters vertically below the top of the hill, and the angle of depression to the car is 25°. Find the height of the hill giving the answer to the nearest meter.

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### Video Transcript

The angle of depression of a car parked on the ground from the top of a hill is 48 degrees. A viewpoint is 14 meters vertically below the top of the hill, and the angle of depression to the car is 25 degrees. Find the height of the hill giving the answer to the nearest meter.

So we’ve been given a lot of information but no diagram. We need to begin by drawing a sketch. We have a hill, and then we have a car parked down on the ground some distance away. We’re told the angle of depression from the top of the hill to the car is 48 degrees. Now, this is where we need to be particularly careful. We sketch in the line of sight between the top of the hill and the car and the horizontal. And then, we remember that the angle of depression is measured down from the horizontal to the line of sight. So it’s this angle here, which is 48 degrees.

We also have a viewpoint on the hill, which is 14 meters vertically below the top of the hill. And then the angle of depression from this point to the car is 25 degrees. Again, we need to be careful. This angle is measured from the horizontal down to the line of sight. So it’s this angle here. So we’ve put all the information in the question onto our diagram. What we’re asked to calculate is the height of the hill. That’s this length here, which we can see is composed of two lengths, the length of 14 meters between the top of the hill and the viewpoint and then a second length, which we’ll need to calculate.

Now, the second length is part of a right triangle formed by the horizontal, the vertical, and the line of sight for the second angle of depression. We can work out one of the angles in this triangle. Between the horizontal and vertical, we have a right angle. So this angle here is 19 minus 25 degrees, which is 65 degrees. We want to calculate the side 𝑥, but in order to do this, we need to know a length in our right triangle. Let’s consider instead the green triangle, a non-right triangle but which has a shared side with the triangle we’re interested in.

We know this triangle has one side of length 14 meters. And we can work out some of the other angles. At the top of the triangle, the angle between the horizontal and vertical is 90 degrees, so the internal angle in the triangle is 90 minus 48, which is 42 degrees. Also internal to the triangle, we have an obtuse angle formed by a right angle and the angle of depression of 25 degrees, so this total angle is 115 degrees. The final angle in this triangle can be calculated using the angle sum in a triangle. 180 minus 42 degrees minus 115 degrees is 23 degrees.

If we know one side length and at least two angles in a non-right triangle, we can apply the law of sines to calculate the length of another side. This tells us that the ratio between a side length and the side of its opposite angle is constant. The side length we want to calculate, which we can call 𝑦 meters, is opposite the angle of 42 degrees. And then we know a side length of 14 meters, which is opposite the angle of 23 degrees. So applying the law of sines, we can say that 𝑦 over sin of 42 degrees is equal to 14 over sin of 23 degrees. We can then multiply both sides of this equation by sin of 42 degrees and evaluate to give 23.975. So we’ve found the shared side.

Returning to our right triangle, we now know one other angle of 65 degrees and the hypotenuse of 23.975. So we can use the cosine ratio to calculate 𝑥. Cosine, remember, is adjacent over hypotenuse, so we have cos of 65 degrees equals 𝑥 over 23.975. We can then multiply both sides by 23.975 and evaluate on our calculators to give 𝑥 equals 10.132. Finally, we must remember we’re looking to calculate the total height of the hill. So that’s the sum of this value and the distance of 14 meters. Adding 14 and then rounding to the nearest meter, we find that the height of the hill is 24 meters.

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