1. **State the problem:** Solve the equation $$\tan \frac{1}{2}x = 3$$ for $$0 < x < 4\pi$$, giving answers in radians to 3 significant figures. 2. **Rewrite the equation:** Let $$\theta = \frac{1}{2}x$$, so the equation becomes $$\tan \theta = 3$$. 3. **Find the general solutions for $$\theta$$:** The tangent function has period $$\pi$$, so $$\theta = \arctan(3) + k\pi, \quad k \in \mathbb{Z}.$$ 4. **Calculate the principal value:** Using a calculator, $$\arctan(3) \approx 1.249$$ radians (to 3 significant figures). 5. **Find $$x$$ in terms of $$\theta$$:** Since $$\theta = \frac{1}{2}x$$, $$x = 2\theta = 2\left(1.249 + k\pi\right) = 2.498 + 2k\pi.$$ 6. **Determine values of $$k$$ such that $$0 < x < 4\pi$$:** - For $$k=0$$: $$x = 2.498$$ (valid) - For $$k=1$$: $$x = 2.498 + 2\pi \approx 2.498 + 6.283 = 8.781$$ (not valid since $$8.781 > 4\pi \approx 12.566$$, but check carefully) Actually, $$4\pi \approx 12.566$$, so $$8.781 < 12.566$$, so $$k=1$$ is valid. - For $$k=2$$: $$x = 2.498 + 4\pi \approx 2.498 + 12.566 = 15.064$$ (not valid) 7. **Also consider the second solution for tangent:** Since $$\tan \theta = 3$$, the solutions are $$\theta = \arctan(3) + k\pi$$, so the two distinct solutions in one period are: - $$\theta_1 = 1.249$$ - $$\theta_2 = 1.249 + \pi = 1.249 + 3.142 = 4.391$$ Corresponding $$x$$ values: - $$x_1 = 2 \times 1.249 = 2.498$$ - $$x_2 = 2 \times 4.391 = 8.782$$ 8. **Check for $$k=0$$ and $$k=1$$:** - For $$k=0$$, $$x = 2.498$$ and $$8.782$$ (both valid) - For $$k=1$$, add $$2\pi$$ to each: - $$2.498 + 2\pi = 2.498 + 6.283 = 8.781$$ (already counted) - $$8.782 + 2\pi = 8.782 + 6.283 = 15.065$$ (exceeds $$4\pi$$) So only $$k=0$$ solutions are valid. 9. **Final answers:** $$x \approx 2.50, 8.78$$ radians (to 3 significant figures) within $$0 < x < 4\pi$$.