How do you find the angles that satisfy sin(x) = -0.5?

To find angles that satisfy sin(x) = -0.5, use the unit circle and sine function properties.

First, recall that the sine function is periodic with a period of 360° (or 2π radians). This means that the sine of an angle repeats every 360°. The sine function is also symmetrical about 180° (or π radians), which helps us find all possible solutions.

The sine of an angle is -0.5 at specific points on the unit circle. The primary angle where sin(x) = -0.5 is 210° (or 7π/6 radians) in the third quadrant. Another angle where sin(x) = -0.5 is 330° (or 11π/6 radians) in the fourth quadrant. These are the principal solutions within one full cycle of 360°.

To find all possible solutions, we use the periodic nature of the sine function. For any integer k, the general solutions can be written as:
x = 210° + 360°k or x = 330° + 360°k (in degrees)
or
x = 7π/6 + 2πk or x = 11π/6 + 2πk (in radians)

Here, k is any integer, which accounts for the infinite number of cycles the sine function can go through. By adding multiples of 360° (or 2π radians), we can find all angles that satisfy sin(x) = -0.5.