Questions: Find two values of (theta, 0 leq theta<2 pi), that satisfy the following equation. [ cos theta=-fracsqrt22 theta=square ] (Simplify your answer. Type an exact answer, using (pi) as needed. Use integers or fractions for any numbers as needed.)

Solution Steps

To find the values of $\theta$ that satisfy $\cos \theta = -\frac{\sqrt{2}}{2}$, we need to identify the angles in the unit circle where the cosine value is $-\frac{\sqrt{2}}{2}$. These angles are typically found in the second and third quadrants. The reference angle for $\cos \theta = \frac{\sqrt{2}}{2}$ is $\frac{\pi}{4}$. Therefore, the angles in the specified range are $\theta = \frac{3\pi}{4}$ and $\theta = \frac{5\pi}{4}$.

To solve the equation $\cos \theta = -\frac{\sqrt{2}}{2}$, we recognize that the cosine function is negative in the second and third quadrants. The reference angle corresponding to $\frac{\sqrt{2}}{2}$ is $\frac{\pi}{4}$.

Using the reference angle, we find the angles in the specified range $0 \leq \theta < 2\pi$:

• In the second quadrant: $$\theta_1 = \pi - \frac{\pi}{4} = \frac{3\pi}{4}$$
• In the third quadrant: $$\theta_2 = \pi + \frac{\pi}{4} = \frac{5\pi}{4}$$

Calculating the decimal values of the angles:

• $\theta_1 = \frac{3\pi}{4} \approx 2.3562$
• $\theta_2 = \frac{5\pi}{4} \approx 3.9270$

Final Answer