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And here is your shmoop de jour...

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Find the cosine 105 degrees.

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And here are your options...Hm, great question.

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Alright let's use the difference formula for cosine.

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Cosine times A minus B equals cosine

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A times cosine B plus sine A

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time sine B. Now have to decide what the values A and B are.

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What are two friendly angles?

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As in nice trig values that have a different of 105

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150 - 45 is 105, though A equals 150 and B equals

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45. Now I can plug it into the difference formula for cosine.

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We can figure out what cosine of 150 degrees is based on the cosine of

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30 degrees. The 150 degrees is just a mulitple of 30 degrees

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but in the second quadrant or cosine is negative...the cosine of 150

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equals negative cosine 30 degrees. Cosine of 30 degrees equal square root of

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3 over 2

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so cosine to 150 degrees is negative root 3

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over 2. Cosine of 45 degrees is the square root of two over two.

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We can find sine of 150 degrees based on sine of

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30 degrees because once again a 150 is a multiple of 30 degrees

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since sin is still positive in Quadrant 2...

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...Sine 150 degrees equals the same value sine of thirty degrees.

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Sine 30 is one half and so is sin 150

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Sine 45 is the square root of 2 over 2

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we have negative square root three over two times the square root

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2 over two plus one half time through 2 over 2

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Simplified we get negative square root of six over four-plus the square root of

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two over 4

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which equals root 2 minus root 6 all over four.

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Option C is our answer.

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