Find the values of x that satisfy the equation 3sin(x) - 1 = 0.
Correct Answer: x = sin^{-1}(1/3) + 2nπ, x = π - sin^{-1}(1/3) + 2nπ, n ∈ Z
- Step 1: Start with the equation 3sin(x) - 1 = 0.
- Step 2: Add 1 to both sides of the equation to isolate the term with sin(x).
- Step 3: The equation now looks like 3sin(x) = 1.
- Step 4: Divide both sides by 3 to solve for sin(x).
- Step 5: Now the equation is sin(x) = 1/3.
- Step 6: Use the inverse sine function (arcsin) to find the angle x that satisfies this equation.
- Step 7: Remember that sine is positive in the first and second quadrants, so find both solutions in the specified interval.
- Trigonometric Equations – The question tests the ability to solve basic trigonometric equations, specifically involving the sine function.
- Inverse Trigonometric Functions – Understanding how to find angles corresponding to specific sine values using inverse functions.
- Periodic Nature of Sine – Recognizing that sine is a periodic function and identifying all possible solutions within a given interval.
