Question:

Step 1: Find the value of $\tan A$ from the given equation. We are given that $3\tan A + 4 = 0$. Solving for $\tan A$: $$\tan A = -\frac{4}{3}$$ Step 2: Determine the signs of $\sin A$ and $\cos A$ using the quadrant information. Since $A$ lies in the fourth quadrant, we know that: - $\sin A < 0$ (sine is negative in the fourth quadrant) - $\cos A > 0$ (cosine is positive in the fourth quadrant) Step 3: Find the exact values of $\sin A$ and $\cos A$. From $\tan A = -\frac{4}{3}$, we can use the identity $\tan A = \frac{\sin A}{\cos A}$. Consider a right triangle where the opposite side is $-4$ and the adjacent side is $3$. The hypotenuse is: $$\text{hypotenuse} = \sqrt{(-4)^2 + 3^2} = \sqrt{16 + 9} = \sqrt{25} = 5$$ Therefore: $$\sin A = -\frac{4}{5}, \quad \cos A = \frac{3}{5}$$ (Note: $\sin A$ is negative and $\cos A$ is positive, consistent with the fourth quadrant.) Step 4: Calculate $\sin 2A$ using the double angle formula. Using the formula $\sin 2A = 2\sin A \cos A$: $$\sin 2A = 2 \cdot \left(-\frac{4}{5}\right) \cdot \frac{3}{5} = 2 \cdot \left(-\frac{12}{25}\right) = -\frac{24}{25}$$ Step 5: Substitute all values into the expression $5\sin 2A + 2\sin A + 4\cos A$. $$5\sin 2A + 2\sin A + 4\cos A = 5 \cdot \left(-\frac{24}{25}\right) + 2 \cdot \left(-\frac{4}{5}\right) + 4 \cdot \frac{3}{5}$$ $$= -\frac{120}{25} - \frac{8}{5} + \frac{12}{5}$$ Step 6: Simplify the expression by converting to a common denominator. Converting all terms to denominator 5: $$= -\frac{24}{5} - \frac{8}{5} + \frac{12}{5}$$ $$= \frac{-24 - 8 + 12}{5} = \frac{-20}{5} = -4$$ **Final Answer:** The value of $5\sin 2A + 2\sin A + 4\cos A$ is $\boxed{-4}$, which corresponds to **Option 4**.