Intuitive Trigonometry/Appendix 1: Solutions to Problems

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.