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A 90-foot-tall building has a shadow that is two feet long.
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What is the angle of elevation of the Sun?
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Let’s begin by sketching a diagram of this scenario.
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Remember, a sketch doesn’t need to be to scale, but it should be roughly in proportion so we can check the suitability of any answers we get.
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The building has a height of 90 feet and its shadow is two feet long.
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We can assume that the angle between the building and its shadow is 90 degrees.
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We’re looking to find the angle of elevation of the Sun.
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In this case, that’s the angle between the horizontal and the line made between the end of the shadow and the top of the building.
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It’s this angle 𝜃.
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So we have a right-angled triangle with two known lengths, in which we’re trying to find an angle 𝜃.
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We need to use right angle trigonometry to do this.
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We’re gonna start by labelling the sides of the triangle.
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The hypotenuse is the longest side.
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It’s the side situated directly opposite the right angle.
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The opposite is the side opposite the given angle.
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It’s the one furthest away from the angle 𝜃.
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Finally, the adjacent side is the other side.
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It’s located next to the angle 𝜃.
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We can see that we know both the length of the opposite and the adjacent side.
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This means we need to use the tan ratio.
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Tan 𝜃 is equal to opposite over adjacent.
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Substituting the values from our triangle into the formula gives us tan of 𝜃 is equal to 90 divided by two, which is 45.
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To solve this equation, we’ll find the inverse tan of both sides.
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The inverse tan of tan 𝜃 is simply 𝜃, so 𝜃 is equal to the inverse tan of 45.
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Inverse tan of 45 is 88.7269.
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Correct to two decimal places, the angle of elevation of the Sun is 88.73 degrees.