Question

ifcottheta =4/3 and theta is acute show that 3sintheta +4costheta=5​

Answer 1

$\cot( \theta) = \frac{4}{3} \\ \\ \tan( \theta) = \frac{3}{4}$

Now,

From basic Trigonometric ratios we know that :-

$\tan(x) = \frac{p}{b}$

Thus we can say that,

For given right Triangle :-

Perpendicular = 3k

Base = 4k

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Applying Pythagoras Theorem :-

H²=P²+B²

H² = 9k² + 16k² = 25k²

H = 5k

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We know that :-

$\sin(x) = \frac{p}{h} \\ \\ \sin( \theta) = \frac{3k}{5k} = \frac{3}{5}$

Similarly :-

$\cos( \theta) = \frac{4}{5}$

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To prove :-

$3 \sin( \theta) + 4 \cos( \theta) = 5$

LEFT HAND SIDE :-

$= 3 \times \frac{3}{5} + 4 \times \frac{4}{5} \\ \\ \\ = \frac{9}{5} + \frac{16}{5} \\ \\ \\ = \frac{25}{5} \\ \\ \\ =5 \\ \\ \\ = rhs$

Hence, Proved.

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