GENERAL SOL.|SINtheta| = 1/2
|sinθ| = 1/2 means that
sinθ = 1/2 or sinθ = -1/2
sinθ = 1/2 in 1st and 2nd quadrants.
sinθ = -1/2 in 3rd and 4th quadrants
So, θ = nπ ± π/6 for any integer n.
To solve the equation |sin(theta)| = 1/2, you need to find the values of theta that make the absolute value of sin(theta) equal to 1/2.
Step 1: Identify the possible values of sin(theta) that satisfy the equation.
The absolute value of a number is always positive, so sin(theta) must be either 1/2 or -1/2.
Step 2: Solve for theta when sin(theta) = 1/2.
To find the values of theta when sin(theta) = 1/2, you can use inverse trigonometric functions. In this case, the inverse sine function (sin^(-1) or arcsin) is used:
theta = sin^(-1)(1/2) + n*2π or theta = π - sin^(-1)(1/2) + n*2π
The inverse sine function returns the angle whose sine value is equal to the given input. Since sin(theta) = 1/2 has multiple solutions, we use n*2π to represent the possibility of obtaining infinitely many angles that satisfy the equation (where n is an integer).
Using a calculator or trigonometric tables, you can determine the values of theta for which sin(theta) = 1/2. The solutions are approximately theta = π/6 + n*2π or theta = 5π/6 + n*2π, where n is an integer.
Step 3: Solve for theta when sin(theta) = -1/2.
To find the values of theta when sin(theta) = -1/2, you can use the inverse sine function again:
theta = sin^(-1)(-1/2) + n*2π or theta = π - sin^(-1)(-1/2) + n*2π
Using a calculator or trigonometric tables, you can determine the values of theta for which sin(theta) = -1/2. The solutions are approximately theta = -π/6 + n*2π or theta = 7π/6 + n*2π, where n is an integer.
Overall, the solutions to the equation |sin(theta)| = 1/2 are:
theta = π/6 + n*2π, 5π/6 + n*2π, -π/6 + n*2π, 7π/6 + n*2π, where n is an integer.