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π΄π΅πΆ is a triangle where π is 96 and the measure of the angle at π΅ is equal to three times the measure of the angle at π΄, which is equal to 90 degrees.
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Find length π giving the answer in terms of sine.
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Itβs always sensible to begin by sketching a diagram out.
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It doesnβt 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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Weβre told that the measure of the angle at π΅ is equal to three times the measure of the angle at π΄ and that these two expressions are equal to 90 degrees.
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This of course means that the measure of the angle at π΅ is 90 degrees.
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Itβs a right angle.
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Letβs look at π΄ then.
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Since three times the measure of the angle π΄ is 90 degrees, we can work out the measure of the angle at π΄ by dividing through by three.
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And doing so, we can see that the measure of the angle at π΄ is 30 degrees.
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We can also work out the measure of the angle at πΆ.
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Angles in a triangle sum to 180 degrees.
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So we can subtract 90 and 30 from 180.
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And that tells us the measure of the angle at πΆ is 60 degrees.
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Now, letβs label the sides of the triangle.
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The side opposite angle π΄ is lowercase π.
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The side opposite angle π΅ is lowercase π.
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And the side opposite angle πΆ is lowercase π.
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We want to calculate the length of π.
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Now normally, we could use right angle trigonometry here.
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However, the question has asked us to calculate it in terms of sine.
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Instead, weβll use the law of sines thatβs π over sin π΄ equals π over sin π΅, which equals π over sin πΆ.
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Or alternatively, sin π΄ over π equals sin π΅ over π, which equals sin πΆ over π.
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Since weβre trying to calculate the length of a side, weβll use the first form.
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In fact, we can actually use either form of this formula.
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However, weβll need to do less rearranging if we use the first one this time.
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Remember, we usually only need to use two parts of this formula.
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Here, we know the length of the side π and weβre trying to find an expression for the side marked π.
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Weβre going to use π over sin π΄ and π over sin πΆ then.
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Letβs substitute the values from our triangle into the formula.
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We get 96 over sin 30 is equal to π over sin 60.
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We want to make π the subject.
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So weβll multiply both sides of this equation by sin 60.
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And that gives us π is equal to 96 over sin 30, all multiplied by sin 60.
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Remember, we can write sin 60 as sin 60 over one.
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And then, when we multiply the numerator of the first fraction by the numerator of the second fraction, we get 96 sin 60.
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And multiplying the two denominators, we get sin 30.
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We were told to leave our answer in terms of sine.
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So weβve finished.
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π is equal to 96 sin 60 over sin 30.