SOLUTION: given that {{{cos(x) = 3/5}}}, {{{3pi/2 < x < 2pi}}} find {{{cos(x/2)}}}, {{{sin(x/2)}}}. {{{tan(x/2)}}}

Algebra ->  Trigonometry-basics -> SOLUTION: given that {{{cos(x) = 3/5}}}, {{{3pi/2 < x < 2pi}}} find {{{cos(x/2)}}}, {{{sin(x/2)}}}. {{{tan(x/2)}}}      Log On


   

Question 921668: given that cos%28x%29+=+3%2F5, 3pi%2F2+%3C+x+%3C+2pi find
cos%28x%2F2%29, sin%28x%2F2%29. tan%28x%2F2%29
Answer by Edwin McCravy(20059) About Me  (Show Source):
You can put this solution on YOUR website!
cos%28x%29+=+3%2F5, 3pi%2F2+%3C+x+%3C+2pi find 
cos%28x%2F2%29, sin%28x%2F2%29, and tan%28x%2F2%29.

Since 3pi%2F2%3Cx%3C2pi, dividing the three sides of the inequality by 2,

3pi%2F4%3Cx%2F2%3Cpi, which tells us that x%2F2 is in quadrant II

To find cos%28x%2F2%29 will need the half-angle formulas:

cos%28x%2F2%29=+%22%22+%2B-+sqrt%28%281%2Bcos%28x%29%29%2F2%29

But since x%2F2 is in quadrant II, its cosine is negative, so we
will only use the negative sign:

cos%28x%2F2%29=+-sqrt%28%281%2Bcos%28x%29%29%2F2%29=+-sqrt%28%281%2B3%2F5%29%2F2%29 

To simplify %281%2B3%2F5%29%2F2, multiply top and bottom by 5, %285%2B3%29%2F10=8%2F10=4%2F5

So 

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To find sin%28x%2F2%29 will need the half-angle formulas:

sin%28x%2F2%29=+%22%22+%2B-+sqrt%28%281-cos%28x%29%29%2F2%29

But since x%2F2 is in quadrant II, its sine is positive, so we
will only use the positive sign:

sin%28x%2F2%29=+-sqrt%28%281-cos%28x%29%29%2F2%29=+-sqrt%28%281-3%2F5%29%2F2%29 

To simplify %281-3%2F5%29%2F2, multiply top and bottom by 5, %285-3%29%2F10=2%2F10=1%2F5

So 

-------------------------

To find tan%28x%2F2%29 there are several half-angle identities we could
use.

 

However I think since I already have the sine and cosine of x%2F2,
I'll use the quotient identity:



Edwin