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Find the value of cos two π΄ given cos of π΄ is negative three-fifths, where π΄ is between 90 degrees and 180 degrees, without using a calculator.
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To find this value, we can use the formula cos two π΄ is equal to two cos squared π΄ minus one.
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And we can plug in the cosine of π΄ as negative three-fifths.
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So we plug in negative three-fifths for the cosine of π΄, and then we will have to square it.
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So if we square negative three, we get nine and if we square five, we get 25.
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So now we take two times nine twenty-fifths.
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So we take two times nine, and then we divide by 25.
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So two times nine is 18, and then we put it over 25 or divide by 25.
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And now we subtract one.
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When we subtract fractions, we need to have common denominators, so we can make one be twenty-five twenty-fifths because that is equal to one.
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So now we can subtract our numerators, and then keep our common denominator of 25.
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And we get negative seven twenty-fifths.
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Now it also told us that π΄ is between 90 degrees and 180 degrees.
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So this will put us in quadrant number two out of the four quadrants because weβre between 90 degrees and 180 degrees.
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So we have π₯, π¦ labelled for a reason.
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Cos π represents the π₯-value, so since weβre in the second quadrant, our π₯-value is negative so our cosine value should be negative.
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And indeed we did; we got a negative seven twenty-fifths.
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So again, our final answer would be negative seven twenty-fifths.