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For the given figure, π΄π΅ is equal to three and π΅πΆ is equal to π.
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Use the law of sines to work out π.
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Give your answer to two decimal places.
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Letβs begin by adding the given measurements to our triangle.
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We can see that we have a nonright-angled triangle for which we know the measure of two angles and the length of one side.
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To find the length of the side labelled π, weβll need to use the law of sines: π over sin π΄ equals π over sin π΅, which equals π over sin πΆ.
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Alternatively, that can be written as sin π΄ over π equals sin π΅ over π, which equals sin πΆ over π.
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We only need to use one of these forms.
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Since weβre trying to calculate the length of one of the sides, weβll use the first form.
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It doesnβt particularly matter either way.
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But by using the first form here, it will minimize the amount of rearranging weβll need to do to solve the equation.
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Next, weβll label the sides of the triangle.
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The side opposite the angle π΄ is already given by the lowercase π.
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The side opposite the angle π΅ is lowercase π, and the side opposite the angle πΆ is lowercase π.
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For the law of sines, we usually only need to use two parts.
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We donβt know the side labelled lowercase π, so weβre going to use π over sin π΄ and π over sin πΆ.
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Letβs substitute what we know into this formula.
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That gives us π over sin 64 is equal to three over sin 31.
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To solve this equation and work out the value of π, weβll need to multiply both sides by sine of 64.
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π is therefore equal to three over sine of 31 multiplied by sine of 64.
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If we put that into our calculator, we get that π is equal to 5.2353 and so on.
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Correct to two decimal places, π is equal to 5.24.
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Notice how there were no units provided in the question, so no units are required in our answer.