### Instruction: Compute $\tan 20^\circ + 4 \sin 20^\circ.$.### Response: To solve this problem, I will try to rewrite the trigonometric expressions in terms of a common function, such as cosine.
I recall that $\tan \theta = \frac{\sin \theta}{\cos \theta}$ and $\sin \theta = \cos (90^\circ - \theta).$
Using these identities, I can rewrite $\tan 20^\circ$ as $\frac{\cos (90^\circ - 20^\circ)}{\cos 20^\circ} = \frac{\cos 70^\circ}{\cos 20^\circ}.$
Similarly, I can rewrite $4 \sin 20^\circ$ as $4 \cos (90^\circ - 20^\circ) = 4 \cos 70^\circ.$
Now I have $\frac{\cos 70^\circ}{\cos 20^\circ} + 4 \cos 70^\circ = \frac{\cos 70^\circ + 4 \cos 20^\circ \cos 70^\circ}{\cos 20^\circ}.$
To simplify this fraction, I will use the product-to-sum identity $\cos \alpha \cos \beta = \frac{1}{2} (\cos (\alpha + \beta) + \cos (\alpha - \beta)).$
Applying this identity to the numerator, I get $\cos 70^\circ + 4 \cos 20^\circ \cos 70^\circ = \cos 70^\circ + 2 (\cos (90^\circ) + \cos (50^\circ)).$
But $\cos 90^\circ = 0,$ so this reduces to $\cos 70^\circ + 2 \cos 50^\circ.$
Now I have $\frac{\cos 70^\circ + 2 \cos 50^\circ}{\cos 20^\circ}.$
To evaluate this fraction, I will use a calculator.
I get $\frac{0.3420201433 + 2 \times 0.6427876097}{0.9396926208} \approx 2.366926456.$