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Question Video: Calculating an Angle of Elevation Using Lengths Mathematics • 11th Grade

Anthony stands 40 m from a building that is 25 m high. What is the angle of elevation from Anthony to the top of the building? Round your answer to the nearest degree.

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Video Transcript

Anthony stands 40 meters from a building that is 25 meters high. What is the angle of elevation from Anthony to the top of the building? Round your answer to the nearest degree.

Let’s begin by sketching the problem. There is a building that is 25 meters high. Anthony is standing 40 meters away. And we can assume that the ground is horizontal and the building is vertical. So these two lines are perpendicular to one another. We are asked to find the angle of elevation from Anthony to the top of the building.

An angle of elevation is the angle measured from the horizontal to the line of sight when we look up towards an object. If we draw in the line of sight from Anthony to the top of the building then, the angle we’re looking for is this angle here. Let’s call that angle 𝜃.

Now, as we have a right triangle in which we know two of the side lengths, we can approach this problem using right triangle trigonometry. We’ll begin by labeling the sides of this triangle, those whose lengths we know, in relation to this angle of 𝜃. The side of length 25 meters is the opposite side, and the side of length 40 meters is the adjacent.

Recalling the acronym SOH CAH TOA then, it is the tangent ratio that we need to use in this problem. This is defined as follows. For an angle 𝜃 in a right triangle, tan of angle 𝜃 is equal to the length of the opposite side divided by the length of the adjacent side. We know the lengths of these two sides, and so we can form an equation. We have tan of 𝜃 is equal to 25 over 40. Now, this fraction can be simplified by dividing both the numerator and denominator by five to give five-eighths.

To find the value of 𝜃, we need to apply the inverse tangent function, giving 𝜃 is equal to the inverse tan of five-eighths. We can now evaluate this on our calculator, which must be in degree mode. It gives 32.005 continuing.

We’re asked to round our answer to the nearest degree. So we’ll round this value to the nearest integer, which is 32. By recalling then that an angle of elevation is the angle formed between the horizontal and the line of sight when we look up towards an object and then applying right triangle trigonometry, we found that the angle of elevation from Anthony to the top of the building is 32 degrees to the nearest degree.

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