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