Question

if sin x cos y=1/4 and 3tan x = 4tan y, then sin(x - y)=1/k .find k

Answer 1

Step-by-step explanation:

We have,

$\sin(x) \cos(y) = \frac{1}{4} \\ 3 \tan(x) = 4 \tan(y)$

Now,

$\frac{ \tan(x) }{ \tan(y) } = \frac{4}{3}$

$\implies \: \frac{ \sin(x) \cos(y)}{ \cos(x)\sin(y) } = \frac{4}{3}$

$\implies \: \frac{ \frac{1}{4} }{ \cos(x)\sin(y) } = \frac{4}{3}$

$\implies \: \frac{ 1}{ 4\cos(x)\sin(y) } = \frac{4}{3}$

$\implies \:\cos(x)\sin(y) = \frac{3}{16}$

Now,

$\sin(x - y) = \frac{1}{k}$

$\implies \: \sin(x) \cos(y) - \cos(x) \sin(y) = \frac{1}{k}$

$\implies \: \frac{1}{4} - \frac{3}{16} = \frac{1}{k}$

$\implies \: \frac{4 - 3}{16} = \frac{1}{k}$

$\implies \: \frac{1}{16} = \frac{1}{k}$

$\implies \: k = 16$