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Expert: - 5/5/2009


Question
QUESTION: I've being asked to reduce the express to a single term_
sinx*cos2x+sin2x*cosx
can you help me

ANSWER: It is known that sin2x = 2 sinx cosx.
It is also known that cos2x = cos²x - sin²x.

This converts the problem to
= sinx(cos²x - sin²x) + (2sinx cosx)cosx
= sinx(cos²x - sin²x + 2cos²x)
= sinx(3cos²x - sin²x).

This can be converted to sinx since we know that cos²x = 1 - sin²x.

Thus, sinx(3cos²x - sin²x)
= sinx(3(1-sin²x) - sin²x)
= 3sinx(3 - 4sin²x).


---------- FOLLOW-UP ----------

QUESTION: thanks, the follow up question is "find sin(x/2) if cosx = 12/13 and is in the 1st quad.

i understand that sin²x = 1-cos(2x) over 2
from which can be put 1-2--12/13 over 2
from there i lose myself, anyone can help me?

Answer
With (1 - cos(2x))/2, that is the same as (1 - 12/13)/2.
Now 1 is 13/13, so when we subtract off 12/13, we get 1/13.

When this fraction is divided by 2,
it is the same as multiplying by 1/2.

It can be seen that (1/2)(1/13) = 1/26.

So what we have is sin²x = 1/26, which means that sinx = √1 / √26.
To put this in proper format, multiply by √26/√26 giving √26 / 26.

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