CBSE Class 12-science Answered

if sin x+2cos x=1 then find Tha value of 2sin x - cosx

Asked by gopalkumarrai87 | 11 Oct, 2023, 11:02: AM

Expert Answer

Let t = tan(x/2)

begin mathsize 14px style sin left parenthesis x right parenthesis space equals space fraction numerator 2 space tan left parenthesis x divided by 2 right parenthesis over denominator 1 plus space tan squared left parenthesis x divided by 2 right parenthesis space end fraction space equals space fraction numerator 2 space t over denominator 1 space plus space t squared end fraction end style

begin mathsize 14px style cos left parenthesis x right parenthesis space equals fraction numerator 1 space minus space tan squared left parenthesis x divided by 2 right parenthesis over denominator 1 space plus space tan squared left parenthesis x divided by 2 right parenthesis end fraction space equals space fraction numerator 1 space minus space t squared over denominator 1 plus t squared end fraction end style

begin mathsize 14px style sin space left parenthesis x right parenthesis space plus space 2 space cos space left parenthesis x right parenthesis space equals space 1 end style

begin mathsize 14px style fraction numerator 2 space t over denominator 1 space plus space t squared end fraction space plus space space 2 cross times fraction numerator 1 space minus space t squared over denominator 1 plus t squared end fraction space equals space 1 end style

begin mathsize 14px style 2 space t space plus space 2 space minus space 2 space t squared space equals space 1 space plus space t squared end style

begin mathsize 14px style 3 space t squared minus space 2 t space minus space 1 space equals space 0 end style

begin mathsize 14px style left parenthesis space t space minus 1 space right parenthesis space left parenthesis space 3 space t space plus space 1 space right parenthesis space equals space 0 end style

Hence t = 1 or t = -1/3

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if t = tan(x/2) = 1 , then x =

begin mathsize 14px style straight pi end style

2 sin(x) - cosx = 2 - 0 = 2

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if t = -1/3 , then

begin mathsize 14px style 2 space sin left parenthesis x right parenthesis space minus space cos space left parenthesis x right parenthesis space equals space 2 space cross times fraction numerator 2 t over denominator 1 plus t squared end fraction minus space fraction numerator 1 space minus space t squared over denominator 1 plus t squared end fraction end style

begin mathsize 14px style R H S space equals space fraction numerator 9 over denominator 10 space end fraction cross times open square brackets negative 4 over 3 minus space 1 space plus 1 over 9 close square brackets space equals space 9 over 10 open square brackets fraction numerator negative 20 over denominator 9 end fraction close square brackets space equals space minus 2 end style

2 sin(x) - cosx = -2

Answered by Thiyagarajan K | 11 Oct, 2023, 11:00: PM

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