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A person in an office building looks out of a window from a height of six meters.
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They see a cat on the sidewalk 10 meters from the building.
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From the person, what is the angle of depression of the cat?
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Assume the sidewalk is horizontal.
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Whenever there is a lot of detail in a question, it can be really useful to draw a diagram to represent the given information.
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Here we’re given the height of the person and the distance the cat is from this building.
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The question also tells us to assume the sidewalk is horizontal, which means that the angle between the sidewalk and the building must be 90 degrees.
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The hypotenuse of this right-angled triangle represents the person’s line of sight.
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We’re interested in this because we’re being asked to find the angle of depression from the person.
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We also know that alternate angles are equal.
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Therefore, we can calculate the angle of depression by calculating the value of 𝜃 in this right-angled triangle.
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Let’s start by labelling our sides.
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We can see that we know the value of both the opposite and the adjacent side.
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We are therefore going to need to use the tan ratio.
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Substituting our given lengths into this ratio gives tan 𝜃 equals six over 10.
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To solve this equation, we need to find the inverse of tan.
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𝜃 is equal to inverse tan of six over 10.
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𝜃 is therefore 30.96 degrees correct to two decimal places.
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The angle of depression of the cat from the person is 30.96 degrees.