If $$ 2 an heta = 3 $$, then what is the value of

$$
\frac{3 \sin heta - 2 \cos heta}{3 \sin heta + 2 \cos heta} \; ?
$$

Question

If $$ 2 	an 	heta = 3 $$, then what is the value of

$$
\frac{3 \sin 	heta - 2 \cos 	heta}{3 \sin 	heta + 2 \cos 	heta} \; ?
$$

Accepted Answer

Here’s the step-by-step solution:

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## Given

If $$ 2 	an 	heta = 3 $$, then what is the value of  
$$
\frac{3 \sin 	heta - 2 \cos 	heta}{3 \sin 	heta + 2 \cos 	heta} \; ?
$$

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## **Step 1: Find $$	an 	heta$$**

$$
2 	an 	heta = 3 \implies 	an 	heta = \frac{3}{2}
$$

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## **Step 2: Express $$\sin 	heta$$ and $$\cos 	heta$$ in terms of $$	an 	heta$$**

We know:
$$
\sin 	heta = \frac{	an 	heta}{\sqrt{1 + 	an^2 	heta}}
$$
$$
\cos 	heta = \frac{1}{\sqrt{1 + 	an^2 	heta}}
$$

Substitute $$	an 	heta = \frac{3}{2}$$:

$$
1 + 	an^2 	heta = 1 + \left(\frac{3}{2}ight)^2 = 1 + \frac{9}{4} = \frac{13}{4}
$$
$$
\sqrt{1 + 	an^2 	heta} = \frac{\sqrt{13}}{2}
$$

So,
$$
\sin 	heta = \frac{\frac{3}{2}}{\frac{\sqrt{13}}{2}} = \frac{3}{\sqrt{13}}
$$
$$
\cos 	heta = \frac{1}{\frac{\sqrt{13}}{2}} = \frac{2}{\sqrt{13}}
$$

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## **Step 3: Substitute into the Expression**

$$
\frac{3 \sin 	heta - 2 \cos 	heta}{3 \sin 	heta + 2 \cos 	heta} = 
\frac{3 \left(\frac{3}{\sqrt{13}}ight) - 2 \left(\frac{2}{\sqrt{13}}ight)}{3 \left(\frac{3}{\sqrt{13}}ight) + 2 \left(\frac{2}{\sqrt{13}}ight)}
$$

$$
= \frac{\frac{9}{\sqrt{13}} - \frac{4}{\sqrt{13}}}{\frac{9}{\sqrt{13}} + \frac{4}{\sqrt{13}}}
$$
$$
= \frac{\frac{5}{\sqrt{13}}}{\frac{13}{\sqrt{13}}}
$$
$$
= \frac{5}{13}
$$

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## **Final Answer**

$$
\boxed{\frac{5}{13}}
$$

Answer

Answer

Answer