Intuitive Trigonometry/Appendix 1: Solutions to Problems

Radians and Arc Length

 * 1) $$\ell_{\text{arc}}=2\pi r\left(\frac{\theta}{360^{\circ}}\right)$$ is our formula for arc length; we can plug in our given values and solve. $$\ell=2\pi\times 2\left(\frac{10^{\circ}}{360^{\circ}}\right)$$ which simplifies to $$\ell=\frac{4\pi}{36}$$ (notice the degree signs cancel) and further simplifies to $$\ell=\frac{\pi}{9}$$ which is our solution.
 * 2) $$2\pi\text{ rad}=360^{\circ}$$, so we can create a proportion
 * 3) $$2\pi\text{ rad}=360^{\circ}$$, so we can create a proportion
 * 4) $$\ell_{\text{arc}}=2\pi r\left(\frac{\theta}{360^{\circ}}\right)$$ is our formula for arc length; we can plug in our given values and solve. $$5=2\pi\times 4\left(\frac{\theta}{360^{\circ}}\right)$$ which simplifies to $$5=\frac{8\pi\theta}{360^{\circ}}$$ or $$1800^{\circ}=8\pi\theta$$ or $$\frac{1800}{8\pi}^{\circ}=\frac{225}{\pi}^{\circ}=\theta$$, which is our solution.