Question

if 3 tan theta = 4,find the value of 4 cos theta - sin theta / 2 cos theta + sin theta​

Answer 1

$3 tan\theta=4\\ \\ \\tan\theta=\dfrac{4}{3}\\ \\ \\ \dfrac{p}{b}=\dfrac{4}{3}\\ \\ \\p=4k\\ \\b=3k$

Hence;

Applying pythagoras Theorem :-

$h^2=p^2+b^2\\ \ \\h^2=16k^2+9k^2\\ \\ \\h^2=25k^2\\ \\ \\h=\sqrt{25k^2} \\ \\ \\h=5k$

Now:-

$sin\theta=\dfrac{p}{h}=\dfrac{4}{5}\\ \\ \\cos\theta=\dfrac{b}{h}=\dfrac{3}{5}$

Now we have :-

$\dfrac{4cos\theta-sin\theta}{2cos\theta+sin\theta}\\ \\ \\=\dfrac{4*\dfrac{3}{5}-\dfrac{4}{5}}{2*\dfrac{3}{5}+\dfrac{4}{5}}\\ \\ \\=\dfrac{\dfrac{12-4}{5}}{\dfrac{6+4}{5}}\\ \\ \\=\dfrac{8}{10}\\ \\ \\=\dfrac{4}{5}$

Answer 2

See the Attachment.

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