Intuitive Trigonometry/Appendix 1: Solutions to Problems

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Appendix 1: Solutions to Problems

Radians and Arc Length

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