Question

PQR is an isosceles triangle whose equal sides PQ and PR are at right angles. S and T are points on PQ such that QS=6SP and QT=2TP.PRS=θ,PRT=ϕ

Answer 1

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Let equal sides be $a$

$\tt tan\theta = \dfrac{SP}{PR} =\dfrac{\frac{a}{2} }{a} =\dfrac{1}{2} \\ \\\\ \\ sin\theta =\dfrac{1}{\sqrt{50} } \\ \\ \\ tan\phi=\dfrac{PT}{TR}=\dfrac{\frac{a}{3} }{a}=\dfrac{1}{3} \\ \\ \\ sin\phi=\dfrac{1}{\sqrt{10} } \\ \\ \\ sin2\phi=\dfrac{2tan\phi}{1+tan^{2}\phi } =\dfrac{2\times\frac{1}{3} }{1+\frac{1}{9} } =\dfrac{3}{5} \\ \\ \\ cos2\phi=\dfrac{4}{5} \\ \\ \\ sin\phi+cos\phi=\dfrac{1}{\sqrt{10} } +\dfrac{3}{\sqrt{10} } =\dfrac{4}{\sqrt{10} }$

$\tt sin(2\phi+\theta )=sin(2\phi)cos2\phi+cos2\phi sin\theta \\ =\dfrac{3}{5} \times\dfrac{3}{\sqrt{50} } +\dfrac{4}{5} \times\dfrac{1}{\sqrt{50} } =\dfrac{1}{\sqrt{2} }$

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