Questions: Find all the angles θ in [0°, 360°) that have a sin(θ) = -√2/2.

Solution Steps

To find all the angles \(\theta\) in the interval \([0^\circ, 360^\circ)\) that have \(\sin(\theta) = \frac{-\sqrt{2}}{2}\), we need to identify the angles where the sine function takes this value. The sine function is negative in the third and fourth quadrants. The reference angle for \(\frac{\sqrt{2}}{2}\) is \(45^\circ\). Therefore, the angles in the specified interval are \(225^\circ\) and \(315^\circ\).

We are given the equation \( \sin(\theta) = \frac{-\sqrt{2}}{2} \).

The reference angle corresponding to \( \frac{\sqrt{2}}{2} \) is \( 45^\circ \).

The sine function is negative in the third and fourth quadrants. Therefore, we need to find the angles in these quadrants.

Using the reference angle:

• In the third quadrant: \[ \theta_1 = 180^\circ + 45^\circ = 225^\circ \]
• In the fourth quadrant: \[ \theta_2 = 360^\circ - 45^\circ = 315^\circ \]

Final Answer