To find the general solution to the equation y = arcsin(0.6428), you need to find all the possible values of y that satisfy the equation. Here's how you can do it:
Step 1: Calculate the principal value of arcsin(0.6428).
The principal value is the first value of y that satisfies the equation. You can use a calculator with inverse trigonometric functions (such as sin^-1) to find the principal value. In this case, arcsin(0.6428) ≈ 40.07°.
Step 2: Find the general solution.
To find the general solution, you need to add or subtract multiples of 360° (or 2π radians) to the principal value. This is because the sine function is periodic with a period of 360°.
So, the general solution for the equation y = arcsin(0.6428) is:
y = 40.07° ± 360°k,
where k is an integer representing all possible values.
In other words, you can find any value of y that satisfies the equation by taking the principal value (40.07°) and adding or subtracting 360° (or any multiple of it) to it. Each time you add or subtract 360°, you get another valid value of y.
In this case, the possible values of y are:
40.07°, 40.07° + 360°, 40.07° - 360°, 40.07° + 2(360°), 40.07° - 2(360°), and so on.
So, the correct answer choices would be:
40°±360°k,
140°±360°k,
220°±360°k,
320°±360°k.
Note: The k in the solution represents an integer (positive, negative, or zero), allowing for all possible values of y to be obtained.