Questions: Solve the equation over the interval [0,2 π). Write numbers using integers or simplified fractions. (a) cos α = -√3/2 (b) cos α = √3/2

$Solve the equation over the interval [0,2 π). Write numbers using integers or simplified fractions. (a) cos α = -√3/2 (b) cos α = √3/2$

Transcript text: Solve the equation over the interval $[0,2 \pi)$. Write numbers using integers or simplified fractions. (a) $\cos \alpha=-\frac{\sqrt{3}}{2}$ (b) $\cos \alpha=\frac{\sqrt{3}}{2}$

Solution

Solution Steps

Step 1: Solve $\cos \alpha = -\frac{\sqrt{3}}{2}$ over $[0, 2\pi)$

The cosine of an angle is $-\frac{\sqrt{3}}{2}$ at $\alpha = \frac{5\pi}{6}$ and $\alpha = \frac{7\pi}{6}$ in the interval $[0, 2\pi)$.

Step 2: Write the solution set for $\cos \alpha = -\frac{\sqrt{3}}{2}$

The solution set is $\left\{\frac{5\pi}{6}, \frac{7\pi}{6}\right\}$.

Step 3: Solve $\cos \alpha = \frac{\sqrt{3}}{2}$ over $[0, 2\pi)$

The cosine of an angle is $\frac{\sqrt{3}}{2}$ at $\alpha = \frac{\pi}{6}$ and $\alpha = \frac{11\pi}{6}$ in the interval $[0, 2\pi)$.

Step 4: Write the solution set for $\cos \alpha = \frac{\sqrt{3}}{2}$

The solution set is $\left\{\frac{\pi}{6}, \frac{11\pi}{6}\right\}$.

Final Answer

(a) $\boxed{\frac{5\pi}{6}, \frac{7\pi}{6}}$

(b) $\boxed{\frac{\pi}{6}, \frac{11\pi}{6}}$

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