Final answer:
Values of θ that satisfy sin(θ) = - √2/2 in the interval [0, 360) are 225° and 315°, which correspond to the 3rd and 4th quadrants where the sine value is negative.
Step-by-step explanation:
To find all values of θ in the interval [0, 360) that satisfy the equation sin(θ) = - √2/2, we need to consider the unit circle and the properties of the sine function. The sine function represents the y-coordinate of a point on the unit circle, and a sine value of - √2/2 corresponds to an angle that lies in the 3rd or 4th quadrant of the unit circle because the sine is negative in these quadrants.
Step 1: Identify the reference angle. The reference angle for sin(θ) = - √2/2 is 45 degrees (or π/4 radians) because sin(45°) = sin(π/4) = √2/2.
Step 2: Determine the angles in the 3rd and 4th quadrants where the sine is negative. These angles are 180° + 45° = 225° (or 5π/4 radians) and 360° - 45° = 315° (or 7π/4 radians).
Therefore, the values of θ in the interval [0, 360) that satisfy sin(θ) = - √2/2 are 225° and 315°.