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Given the following figure, find the lengths π΄πΆ and π΅πΆ and the measure of angle π΄π΅πΆ in degrees.
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Give your answers to two decimal places.
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In order to answer this question, we need to consider the sine, cosine, and tangent trigonometrical ratios.
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sin π is equal to the opposite divided by the hypotenuse.
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cos π is equal to the adjacent divided by the hypotenuse.
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And tan π is equal to the opposite divided by the adjacent.
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Labelling the three sides of the right-angled triangle tells us that π΅πΆ is the hypotenuse, the longest side, π΄π΅ is the opposite as it is opposite the 21-degree angle, and π΄πΆ is the adjacent as it is next to or adjacent to the 90-degree and 21-degree angle.
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The first part of our question was to work out the length of π΄πΆ, labelled π₯ on the diagram.
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π΄πΆ is the adjacent.
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And π΄π΅ is the opposite.
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Therefore, we are going to use the tangent ratio.
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Substituting in these values gives us tan 21 is equal to three divided by π₯.
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Multiplying both sides of the equation by π₯ and then dividing both sides by tan 21 gives us π₯ is equal to three divided by tan 21.
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Typing this into the calculator, gives us an answer for π₯ of 7.82, to two decimal places.
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This means that the length π΄πΆ is equal to 7.82.
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The second part of our question asked us to work out the length of π΅πΆ.
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As we now know π΄πΆ is 7.82 and π΄π΅ is three, we could use Pythagorasβs theorem to work out the length of π΅πΆ.
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However, in this case, weβre going to continue to use the trigonometrical ratios and use the opposite and the hypotenuse to work out the length π΅πΆ.
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Substituting in our values to the sine ratio gives us sin 21 equals three divide by π¦.
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Once again, using the balancing method allows us to swap the π¦ and the sin 21.
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Therefore, π¦ is equal to three divided by sin 21.
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Typing this into the calculator gives us an answer to two decimal places of 8.37.
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This means that the length of π΅πΆ in the triangle is 8.37.
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The last part of our question asked us to work out the angle π΄π΅πΆ, labelled π in the diagram.
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Now we could use our trigonometrical ratios again to calculate π.
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However, we know that angles in a triangle add up to 180 degrees.
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Therefore, 90 degrees plus 21 degrees plus π equals 180 degrees.
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If we write this out as an equation, we can solve it to find the measure of angle π΄π΅πΆ.
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90 plus 21 is 111.
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Therefore, 111 plus π equals 180.
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Subtracting 111 from both sides of the equation, gives us π equal 69 degrees.
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This means that the angle π΄π΅πΆ in the triangle is 69 degrees.
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This question shows that we can use a mixture of our trigonometrical ratios, Pythagorasβs theorem, and our angle properties to work out all the lengths and angles in a right-angled triangle.