SOLUTION: Simplify the following expressions as much as possible. cos(x+pi/3)+sin(x-pi/6)

Algebra ->  Trigonometry-basics -> SOLUTION: Simplify the following expressions as much as possible. cos(x+pi/3)+sin(x-pi/6)      Log On


   

Question 1138453: Simplify the following expressions as much as possible.
cos(x+pi/3)+sin(x-pi/6)
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

Use the identity cos(A+B) = cos(A)*cos(B) - sin(A)*sin(B)

cos%28x%2Bpi%2F3%29+=+cos%28x%29%2Acos%28pi%2F3%29+-+sin%28x%29%2Asin%28pi%2F3%29

cos%28x%2Bpi%2F3%29+=+cos%28x%29%2A%281%2F2%29+-+sin%28x%29%2A%28sqrt%283%29%2F2%29

cos%28x%2Bpi%2F3%29+=+%281%2F2%29%2Acos%28x%29+-+%28sqrt%283%29%2F2%29%2Asin%28x%29

cos%28x%2Bpi%2F3%29+=+%281%2F2%29%2A%28cos%28x%29-sqrt%283%29%2Asin%28x%29%29

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Use the identity sin(A-B) = sin(A)*cos(B) - cos(A)*sin(B)

sin%28x-pi%2F6%29+=+sin%28x%29cos%28pi%2F6%29+-+cos%28x%29sin%28pi%2F6%29

sin%28x-pi%2F6%29+=+sin%28x%29%2A%28sqrt%283%29%2F2%29+-+cos%28x%29%281%2F2%29

sin%28x-pi%2F6%29+=+%28sqrt%283%29%2F2%29%2Asin%28x%29+-+%281%2F2%29%2Acos%28x%29

sin%28x-pi%2F6%29+=+%281%2F2%29%2A%28sqrt%283%29%2Asin%28x%29-cos%28x%29%29

sin%28x-pi%2F6%29+=+%281%2F2%29%2A%28-cos%28x%29%2Bsqrt%283%29%2Asin%28x%29%29

sin%28x-pi%2F6%29+=+-%281%2F2%29%2A%28cos%28x%29-sqrt%283%29%2Asin%28x%29%29

note how this result is basically the negative version of the result from the previous section above.

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cos%28x%2Bpi%2F3%29%2Bsin%28x-pi%2F6%29+=+%281%2F2%29%2Ay-%281%2F2%29%2Ay Let y = cos(x)-sqrt(3)*sin(x)

cos%28x%2Bpi%2F3%29%2Bsin%28x-pi%2F6%29+=+%281%2F2-1%2F2%29%2Ay

cos%28x%2Bpi%2F3%29%2Bsin%28x-pi%2F6%29+=+0%2Ay

cos%28x%2Bpi%2F3%29%2Bsin%28x-pi%2F6%29+=+0

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Answer: The original expression simplifies to 0

We can say that the equation cos%28x%2Bpi%2F3%29%2Bsin%28x-pi%2F6%29+=+0 is an identity, meaning that it is a true equation for all real numbers x.