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.