The value of tan $20^{\circ}+4~sin~20^{\circ}$ is:

Question

Accepted Answer

To determine why Option 1 ($\sqrt{3}$) is the correct answer for the expression $	an 20^{\circ} + 4 \sin 20^{\circ}$, let's analyze the trigonometric values and their relationships.

1. **Understanding the Expression:**
   - We need to evaluate $	an 20^{\circ} + 4 \sin 20^{\circ}$.
   - The tangent and sine functions are related through the identity $	an 	heta = \frac{\sin 	heta}{\cos 	heta}$.

2. **Approximate Values:**
   - While exact values for $	an 20^{\circ}$ and $\sin 20^{\circ}$ are not standard, we can use known approximations or trigonometric identities to estimate them.
   - For small angles, $	an 	heta \approx \sin 	heta \approx 	heta$ in radians. However, $20^{\circ}$ is not small enough for this approximation to be precise.

3. **Trigonometric Identities and Approximations:**
   - Using trigonometric tables or calculators, we find:
     - $	an 20^{\circ} \approx 0.364$
     - $\sin 20^{\circ} \approx 0.342$
   - Therefore, $4 \sin 20^{\circ} \approx 4 	imes 0.342 = 1.368$.

4. **Summing the Values:**
   - Adding these approximations gives:
     - $	an 20^{\circ} + 4 \sin 20^{\circ} \approx 0.364 + 1.368 = 1.732$.
   - Notice that $1.732$ is approximately equal to $\sqrt{3}$, which is about $1.73205$.

5. **Conclusion:**
   - The calculated value of $1.732$ closely matches $\sqrt{3}$, confirming that Option 1 is indeed the correct answer.

6. **Why Other Options are Incorrect:**
   - **Option 2 ($\frac{1}{\sqrt{3}}$):** This value is approximately $0.577$, which is significantly less than the calculated $1.732$.
   - **Option 3 ($\sqrt{2}$):** This value is approximately $1.414$, which is less than $1.732$.
   - **Option 4 ($\frac{1}{\sqrt{2}}$):** This value is approximately $0.707$, which is also much less than $1.732$.

By following these calculations and logical reasoning, we can confidently conclude that Option 1 ($\sqrt{3}$) is the correct answer for the expression $	an 20^{\circ} + 4 \sin 20^{\circ}$.