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πππ is a right-angled triangle at π, where ππ is equal to 16.5 centimeters, ππ is equal to 28 centimeters, and ππ is equal to 32.5 centimeters.
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Find the measure of angle π, giving the answer to the nearest second.
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We will begin by sketching the right triangle πππ.
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We are told that side ππ is 16.5 centimeters long.
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The length of ππ is 28 centimeters.
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And the length of ππ is 32.5 centimeters.
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We are asked to work out the measure of angle π.
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We will do this using our knowledge of right angle trigonometry and the trigonometric ratios, which we can recall using the acronym SOH CAH TOA.
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We recall that the longest side of a right triangle, which is opposite the right angle, is known as the hypotenuse.
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The side opposite angle π is known as the opposite.
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And the side next to angle π and the right angle is known as the adjacent.
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Since we know the lengths of all three sides of our triangle, we can use any one of the three ratios.
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sin π is equal to the opposite over the hypotenuse.
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cos π is equal to the adjacent over the hypotenuse.
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And tan π is equal to the opposite over the adjacent.
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In this question, we will use the sine ratio.
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Substituting in the lengths of the opposite and hypotenuse, we have sin π is equal to 16.5 over 32.5.
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We can then take the inverse sine of both sides of our equation such that π is equal to the inverse sin of 16.5 over 32.5.
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Ensuring that our calculator is in degree mode, we can type in the right-hand side, giving us 30.510237 and so on.
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This is the answer in degrees.
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However, weβre asked to give our answer to the nearest second.
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We will therefore need to convert this value into degrees, minutes, and seconds.
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Since there are 60 minutes in one degree, we multiply the decimal part of our answer by 60.
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This gives us 30.6142 and so on.
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So our angle is equal to 30 degrees and 30.6142 minutes.
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There are 60 seconds in one minute.
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So, once again, we multiply the decimal part of our answer by 60.
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This gives us 36.854 and so on.
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So our angle is equal to 30 degrees, 30 minutes, and 36.854 seconds.
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Rounding this to the nearest second, we have 30 degrees, 30 minutes, and 37 seconds.
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This is the measure of angle π.