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1. Solve $x^2+5x+6=0$.
Solution: The polynomial on the left-hand side factors: $$(x+3)(x+2)=0.$$ Therefore by the zero-product property of the real numbers, we must conclude that $x+3=0$ or $x+2=0$. Thus $x=-3$ or $x=-2$.
2. Convert $32^{\circ}$ to radians.
Solution: Calculate $$32^{\circ} = 32^{\circ} (1) = 32^{\circ} \left( \dfrac{\pi \mathrm{\hspace{2pt} rad}}{180^{\circ}} \right)=\dfrac{32\pi}{180}=\dfrac{8\pi}{45}.$$ 3. Convert $\dfrac{\pi}{8}$ radians to degrees.
Solution: Calculate $$\dfrac{\pi}{8} \mathrm{\hspace{2pt} rad} = \left( \dfrac{\pi}{8} \mathrm{\hspace{2pt} rad} \right) \left( \dfrac{180^{\circ}}{\pi \mathrm{\hspace{2pt} rad}} \right) = \left( \dfrac{45}{2} \right)^{\circ}.$$ 4. What is the arc length subtende by an angle of $\dfrac{\pi}{3}$ radians of a circle with radius $7$?
Solution: In the formula $s=r\theta$, we were told $\theta = \dfrac{\pi}{3}$ and $r=7$. Substituting these values in yields
$$s=\left(7\right) \left( \dfrac{\pi}{3} \right) = \dfrac{7\pi}{3}.$$ 5. What radius must a circle have if an angle of $13^{\circ}$ subtends an arc length of $27$?
Solution: In the formula $s=r\theta$, we are told that $s=27$ and $\theta=13^{\circ}$. But the formula $s=r\theta$ only works when $\theta$ is in radians, so we must convert $13^{\circ}$ to radians in order to use it. So, compute $$13^{\circ} = \left( 13^{\circ} \right) \left( \dfrac{\pi \mathrm{\hspace{2pt} rad}}{180^{\circ}} \right) = \dfrac{13\pi}{180} \mathrm{rad}.$$ Now we plug these into $s=r\theta$ to get $27=r \left( \dfrac{13\pi}{180} \right)$. Solve for $r$ to get $r = \dfrac{27(180)}{13\pi}=\dfrac{4860}{13\pi}$.