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Find the simplified form of $cos^{-1}(\frac{3}{5}cos x+\frac{4}{5}sin x),$where $x\in (\frac{-3}{4},\frac{3}{4})$

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Toolbox:
  • \( cosA\: cosB+sinA\: sinB=cos(A-B)\)
  • \(sinx=\sqrt{1-cos^2x}\)
Let \( \frac{3}{5}=cos\theta\)
\( \Rightarrow\:sin\theta=\frac{4}{5}\)
\(\Rightarrow\:\frac{3}{5}cosx-\frac{4}{5}sinx=cos\theta cosx+sin\theta sinx\)
By taking A=\(\theta\) and B=x in the above formula
\(cos\theta cosx+sin\theta sinx=cos(\theta-x)\)
The given expression becomes
\( cos^{-1} [cos\theta cosx+sin\theta sinx]\)
\( = cos^{-1}\: cos(\theta-x)=\theta-x\)
Substituting the value of \(\theta\), we get
\( =cos^{-1}\frac{3}{5}-x\)
answered Mar 2, 2013 by thanvigandhi_1
edited Mar 7, 2013 by rvidyagovindarajan_1
 
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