Question

If $\sin A = \frac{4}{5}$, then $(3 − \tan A)(2 + \cos A)$ is:

• A. $\frac{12}{5}$
• B. $\frac{13}{3}$
• C. $\frac{13}{5}$
• D. 3

Answer 1

The Correct Option is B

Given: $\sin A = \frac{4}{5}$

First, we'll find $\cos A$ using the Pythagorean identity:

$$\sin^2 A + \cos^2 A = 1 \\ \left(\frac{4}{5}\right)^2 + \cos^2 A = 1 \\ \frac{16}{25} + \cos^2 A = 1 \\ \cos^2 A = 1 - \frac{16}{25} = \frac{9}{25} \\ \cos A = \frac{3}{5} \quad (\text{We take the positive value})$$

Next, we find $\tan A$:

$$\tan A = \frac{\sin A}{\cos A} = \frac{\frac{4}{5}}{\frac{3}{5}} = \frac{4}{3}$$

Now, we'll evaluate the expression $(3 - \tan A)(2 + \cos A)$:

$$(3 - \tan A)(2 + \cos A) = \left(3 - \frac{4}{3}\right)\left(2 + \frac{3}{5}\right) \\ = \left(\frac{9}{3} - \frac{4}{3}\right)\left(\frac{10}{5} + \frac{3}{5}\right) \\ = \left(\frac{5}{3}\right)\left(\frac{13}{5}\right) \\ = \frac{5 \times 13}{3 \times 5} = \frac{13}{3}$$

The correct answer is option (2) $\frac{13}{3}$.

Answer 2

Using $\sin A = \frac{4}{5}$, 

we calculate $\tan A = \frac{4}{3}$ and $\cos A = \frac{3}{5}$. 

Substituting into the expression gives $\left(3 − \frac{4}{3}\right)\left(2 + \frac{3}{5}\right) = \left(\frac{5}{3}\right) \times \left(\frac{13}{5}\right) = \frac{13}{3}$.