Question

Use the space provided to solve

sin (x-2) + 1=1/2 over the interval [0, 2π). Show your work
use basic trigonometric equations ​

Answer 1

The solutions to the trigonometric equation sin(x - 2) + 1 = 1/2 over the interval [0, 2π) are approximately x ≈ 3.7854 and x ≈ 5.4978 when calculated step by step, following the sine function properties.

Step-by-step explanation:

To solve the trigonometric equation sin(x - 2) + 1 = 1/2 over the interval [0, 2π), first isolate the sine function:

sin(x - 2) = 1/2 - 1

sin(x - 2) = -1/2

Now, recall that the sine function oscillates between +1 and -1, and the general solutions for sin(θ) = -1/2 are θ = 7π/6 + 2πn or θ = 11π/6 + 2πn, where n is an integer. Since we need to solve for x in sin(x - 2), substitute back:

x - 2 = 7π/6 + 2πn or x - 2 = 11π/6 + 2πn

Solving for x gives:

x = 7π/6 + 2 + 2πn or x = 11π/6 + 2 + 2πn

The interval for this problem is from 0 to 2π, so we need to identify the n values that keep x within this range. By testing integers, we find:

x = 7π/6 + 2 (for n = 0) and x = 11π/6 + 2 (for n = 0)

These are the solutions within the specified range:

x ≈ 3.7854 and x ≈ 5.4978