Question

if tan theta =1/2 evaluate 2sin theta +3cos TV theta / 4cos theta +3 sin theta
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Answer 1

EXPLANATION.

tan∅ = 1/2.

As we know that,

By using Pythagorean theorem, we get.

⇒ H² = P² + B².

tan∅ = Perpendicular/Base.

⇒ tan∅ = 1/2 = p/b.

⇒ H² = (1)² + (2)².

⇒ H² = 1 + 4.

⇒ H² = 5.

⇒ H = √5.

Sin∅ = Perpendicular/hypotenuse = 1/√5.

Cos∅ = Base/Hypotenuse = 2/√5.

tan∅ = perpendicular/base = 1/2.

cosec∅ = hypotenuse/perpendicular = √5/1.

sec∅ = hypotenuse/base = √5/2.

cot∅ = base/perpendicular = 2/1.

$\sf \implies \dfrac{2Sin \theta + 3Cos \theta}{4Cos \theta + 3Sin \theta}$

$\sf \implies \dfrac{2 \times \dfrac{1}{\sqrt{5} } + 3 \times \dfrac{2}{\sqrt{5} } }{4 \times \dfrac{2}{\sqrt{5}} + 3 \times \dfrac{1}{\sqrt{5} } }$

$\sf \implies \dfrac{\dfrac{2}{\sqrt{5} } + \dfrac{3(2)}{\sqrt{5} } }{\dfrac{8}{\sqrt{5} } + \dfrac{3}{\sqrt{5} } }$

$\sf \implies \dfrac{8 }{11}$

Answer 2

$\large\underline{ \underline{ \sf \maltese{ \: Correct \: Question:- }}}$

$\sf{{If \:tan \theta=}{\dfrac{1}{2}} \:then \:evaluate: {\dfrac{2sin\theta+3cos \theta}{4cos \theta+3sin \theta}}}$

$\large\underline{ \underline{ \sf \maltese{ \: Solution:- }}}$

Now, Simply, dividing the given fraction by cosθ

$\sf{\implies}{\dfrac{{\dfrac{2sin\theta +3cos \theta}{cos \theta}}}{{\dfrac{4cos \theta +3sin \theta}{cos \theta}}}}$

$\sf{\implies}{\dfrac{{\dfrac{2sin\theta}{cos\theta}}+{\dfrac{3cos\theta}{cos\theta}}}{{\dfrac{4cos\theta}{cos\theta}+{\dfrac{3sin\theta}{cos\theta}}}}$

$\sf{\implies}{\dfrac{2tan\theta+3}{4+3tan\theta}}$

Here, we are given that, tanθ= 1/2

$\sf{\implies}{\dfrac{2\times{\dfrac{1}{2}}+3}{4+3\times{\dfrac{1}{2}}}}$

$\sf{\implies}{\dfrac{4}{4+1.5}}$

$\sf{\implies}{\red{\dfrac{4}{5.5}}}$

Multiplying the fraction by 2/2 we get:

$\sf{{\dfrac{4}{5.5}}\times{\dfrac{2}{2}}}$

$\sf{{\implies}{\dfrac{8}{11}}$

So, final answer,

$\sf{\implies}{\dfrac{8}{11}}$