Final answer:
The solutions to the equation sin(x) = 0.35 are x = π/6 + 2nπ and x = 5π/6 + 2nπ. These correspond to angles in the first and second quadrants, where the sine function has a value of 0.35. The other options given do not represent accurate solutions to the equation.
Step-by-step explanation:
The equation we are solving is sin(x) = 0.35. To find the solutions, we must look for values of x where the sine function has the value of 0.35. Sine values repeat every 2π radians, which corresponds to the function's period. Since the sine function is positive in the first and second quadrants, we need to find the reference angle whose sine is 0.35 and then identify the angles in those quadrants that have that sine value.
Using a calculator, we find that the reference angle that corresponds to sin(x) = 0.35 is approximately x₁ = 0.36 radians. In the first quadrant, this reference angle itself is the solution. In the second quadrant, the solution is π - x₁ to ensure the sine value remains positive. Therefore, the general solutions for the equation are:
- x = x₁ + 2nπ
- x = (π - x₁) + 2nπ
Substituting the approximate value of the reference angle, the answers become:
- x = 0.36 + 2nπ
- x = (π - 0.36) + 2nπ
To match the solutions to the given options, we convert 0.36 radians to degrees and find that it is roughly equivalent to π/6 radians. Also, π - x₁ is approximately equal to 5π/6 radians. The correct solutions in terms of the provided options are therefore:
- x = π/6 + 2nπ (Option a)
- x = 5π/6 + 2nπ (Option b)
Note that the other options (c and d) do not correspond to the sine value of 0.35 and therefore are not correct.