Final answer:
To solve sin(k) = 1/2, the equation's solutions can be described as angles k equal to π/6 + 2πn or 5π/6 + 2πn, where n is any integer, and the units for k should be in radians.
Step-by-step explanation:
To solve the equation sin(k) = 1/2, we need to find all the angles k, which when plugged into the sine function, will yield a result of 1/2. Considering the unit circle and the symmetry of the sine function, we can determine that k must correspond to the angles where the y-coordinates are 1/2.
The principal angle that satisfies sin(k) = 1/2 is π/6 (or 30 degrees). However, sine is also 1/2 at 5π/6 (or 150 degrees). Since the sine function repeats every 2π radians, the general solutions can be written using the following formulas:
- k = π/6 + 2πn, where n is any integer.
- k = 5π/6 + 2πn, where n is any integer.
Therefore, all solutions of the equation can be described as k being any angle of the form π/6 + 2πn or 5π/6 + 2πn, where n is any integer. Be sure to use units of radians for angles when solving trigonometric equations to avoid confusion with degrees and to ensure the solution is accurate and reasonable.