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In the given figure, find an expression for the height β in terms of π, π, and π.
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Find an expression for the area of the triangle in terms of π, π, and π.
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Letβs begin by finding an expression for the height β in our triangle.
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If we look carefully, we can partition this triangle somewhat.
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And in doing so, we create a smaller triangle with an angle of π degrees and a length of π.
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Now in fact, this length π represents the length of the hypotenuse of this right-angle triangle.
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And we know that itβs a right-angle triangle because angles on a straight line sum to 180 degrees.
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And since we have an angle of 90 degrees, we can find the angle on the other side of the dotted line by subtracting 90 from 180.
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That gives us 90 degrees.
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And therefore, we have a right-angle triangle.
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And since π is the longest side in the triangle, itβs the side opposite the right angle we know itβs a hypotenuse.
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We can also say that the side in this triangle labelled as lowercase β is the opposite side of the triangle.
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Itβs the side opposite the included angle π degrees.
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And when we notice that we know the length of the opposite and the hypotenuse, we can see that we need to use the sine ratio, where sin of π is equal to opposite over hypotenuse.
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Substituting what we know about our right-angle triangle into this formula, we see that sin π is equal to β, which is the opposite, over the hypotenuse, which is π.
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So sin π is equal to β over π.
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And since the aim of this question is to find an expression for the height β, we need to make β the subject.
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And we do so by multiplying both sides of our equation by π.
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β over π multiplied by π is simply β.
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And sin π multiplied by π is written π sin π.
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And we see that in fact this question was a little bit misleading.
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An expression for the height β in terms of π, π, and π doesnβt actually include π.
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Itβs simply π sin π.
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Next, weβre going to find an expression for the area of the triangle in terms of π, π, and π.
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This time, we recall the formula for area of a triangle.
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Itβs a half multiplied by its base multiplied by its perpendicular height.
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The base of our triangle is π units.
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And the perpendicular height is β.
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So we could say that the area is a half πβ.
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But of course, weβre looking to find an expression for the area of the triangle in terms of π, π, and π.
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We just saw that β is equal to π sin π.
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So we can see that the area is a half π multiplied by π sin π.
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And if we fully simplify this expression, the area of our triangle is a half ππ sin π.