Question

If $\sin \theta = \frac{12}{13}$ then the value of $\frac{2 \cos \theta + 3 \tan \theta}{\sin \theta + \tan \theta + \sin \theta}$ is

• A. $\frac{12}{5}$
• B. $\frac{5}{13}$
• C. $\frac{259}{102}$
• D. $\frac{259}{65}$

Answer 1

No option matches the calculated value

Answer 2

To find the value of the expression $\frac{2 \cos \theta + 3 \tan \theta}{\sin \theta + \tan \theta + \sin \theta}$, given $\sin \theta = \frac{12}{13}$:

Step 1: Simplify the denominator.

The denominator is $\sin \theta + \tan \theta + \sin \theta$. Combining the $\sin \theta$ terms, we get $2 \sin \theta + \tan \theta$. So the expression becomes $\frac{2 \cos \theta + 3 \tan \theta}{2 \sin \theta + \tan \theta}$.

Step 2: Find the values of $\cos \theta$ and $\tan \theta$.

We are given $\sin \theta = \frac{12}{13}$. Using the Pythagorean identity $\sin^2 \theta + \cos^2 \theta = 1$: $\cos^2 \theta = 1 - \sin^2 \theta$ $\cos^2 \theta = 1 - \left(\frac{12}{13}\right)^2$ $\cos^2 \theta = 1 - \frac{144}{169}$ $\cos^2 \theta = \frac{169 - 144}{169}$ $\cos^2 \theta = \frac{25}{169}$ Taking the square root, $\cos \theta = \pm \sqrt{\frac{25}{169}} = \pm \frac{5}{13}$. Since no quadrant is specified, we assume $\theta$ is in the first quadrant where all trigonometric ratios are positive. So, $\cos \theta = \frac{5}{13}$.

Now, calculate $\tan \theta$: $\tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{12/13}{5/13} = \frac{12}{5}$.

Step 3: Substitute the values into the expression.

Numerator: $2 \cos \theta + 3 \tan \theta = 2 \left(\frac{5}{13}\right) + 3 \left(\frac{12}{5}\right)$ $= \frac{10}{13} + \frac{36}{5}$ To add these fractions, find a common denominator, which is $13 \times 5 = 65$: $= \frac{10 \times 5}{65} + \frac{36 \times 13}{65}$ $= \frac{50}{65} + \frac{468}{65}$ $= \frac{50 + 468}{65} = \frac{518}{65}$.

Denominator: $2 \sin \theta + \tan \theta = 2 \left(\frac{12}{13}\right) + \frac{12}{5}$ $= \frac{24}{13} + \frac{12}{5}$ To add these fractions, find a common denominator, which is $13 \times 5 = 65$: $= \frac{24 \times 5}{65} + \frac{12 \times 13}{65}$ $= \frac{120}{65} + \frac{156}{65}$ $= \frac{120 + 156}{65} = \frac{276}{65}$.

Step 4: Calculate the final value of the expression.

Value of expression = $\frac{\text{Numerator}}{\text{Denominator}} = \frac{518/65}{276/65} = \frac{518}{276}$.

Step 5: Simplify the fraction.

Both the numerator and denominator are divisible by 2: $\frac{518 \div 2}{276 \div 2} = \frac{259}{138}$.

Conclusion:

The calculated value of the expression is $\frac{259}{138}$. Comparing this result with the given options: A) $\frac{12}{5}$ B) $\frac{5}{13}$ C) $\frac{259}{102}$ D) $\frac{259}{65}$

None of the provided options match the calculated value of $\frac{259}{138}$. This indicates a potential error in the question or the options provided.