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A five-meter ladder is leaning against a vertical wall such that its base is two meters from the wall.
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Work out the angle between the ladder and the floor, giving your answer to two decimal places.
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We haven’t been given a diagram to accompany this question.
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So the first step is going to be to draw one.
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We have a ladder leaning against a vertical wall.
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The triangle formed by the ladder, the floor, and the wall is a right triangle.
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And in fact, that’s all we really need to draw for our diagram.
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The ladder is five meters long, and its base is two meters away from the wall.
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We are asked to work out the angle between the ladder and the floor.
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So that’s this angle here, which we’ll call 𝑥.
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We have a right triangle in which we know the lengths of two sides.
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And we want to calculate the measure of an angle.
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So we can apply trigonometry.
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We begin by labeling the three sides of the triangle in relation to the angle 𝑥.
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So we have the opposite, the adjacent, and the hypotenuse.
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We then recall the acronym SOHCAHTOA to help us decide which of the three trigonometric ratios to use in this problem.
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The side lengths we know are the adjacent and the hypotenuse.
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So we’re going to be using the cosine ratio.
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This is defined for an angle 𝜃 as the length of the adjacent divided by the length of the hypotenuse.
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Substituting two for the adjacent and five for the hypotenuse, we have that cos of 𝑥 is equal to two-fifths.
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To find the value of 𝑥, we need to apply the inverse cosine function, giving 𝑥 is equal to the inverse cos of two-fifths.
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Evaluating this on our calculators, which must be in degree mode, gives 66.421.
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Finally, we round our answer to two decimal places.
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And we found that the angle between the ladder and the floor is 66.42 degrees.