Solve each equation for exact solutions over the interval [0, 2π).
2sin x + 3 = 4

Verified step by step guidance

1. Start by isolating the sine term in the equation: \(2\sin x + 3 = 4\). Subtract 3 from both sides to get \(2\sin x = 1\).
2. Next, divide both sides of the equation by 2 to solve for \(\sin x\): \(\sin x = \frac{1}{2}\).
3. Recall the unit circle values where \(\sin x = \frac{1}{2}\). Identify the angles in the interval \([0, 2\pi)\) where this is true.
4. The sine function equals \(\frac{1}{2}\) at two points in the interval \([0, 2\pi)\): one in the first quadrant and one in the second quadrant. Write these angles using their exact radian measures.
5. List the exact solutions for \(x\) in the interval \([0, 2\pi)\) based on the angles found in the previous step.

Key Concepts

Solving Basic Trigonometric Equations

This involves isolating the trigonometric function (e.g., sine, cosine) to find the angle values that satisfy the equation. For example, rearranging 2sin x + 3 = 4 to sin x = 1/2 allows us to find x by considering the sine function's values.

Unit Circle and Reference Angles

The unit circle helps determine exact angle solutions for trigonometric functions within a given interval. Knowing the reference angles where sine equals a specific value, and their corresponding quadrants, is essential to find all solutions in [0, 2π).

Interval Restriction and Multiple Solutions

Trigonometric equations often have multiple solutions within a specified interval. Understanding how to find all solutions between 0 and 2π requires knowledge of the periodicity of sine and how to express all valid angles that satisfy the equation.