Solve each equation in x over the interval [0, 2π) and each equation in θ over the interval [0°, 360°). Give exact solutions.
sin (x/2) = √2 ― sin (x/2)

Verified step by step guidance

1. Start by rewriting the given equation: \(\sin\left(\frac{x}{2}\right) = \sqrt{2} - \sin\left(\frac{x}{2}\right)\).
2. Add \(\sin\left(\frac{x}{2}\right)\) to both sides to combine like terms: \(\sin\left(\frac{x}{2}\right) + \sin\left(\frac{x}{2}\right) = \sqrt{2}\), which simplifies to \(2 \sin\left(\frac{x}{2}\right) = \sqrt{2}\).
3. Divide both sides by 2 to isolate the sine term: \(\sin\left(\frac{x}{2}\right) = \frac{\sqrt{2}}{2}\).
4. Recall that \(\sin(\alpha) = \frac{\sqrt{2}}{2}\) at angles \(\alpha = \frac{\pi}{4}\) and \(\alpha = \frac{3\pi}{4}\) within the interval \([0, 2\pi)\), and similarly for degrees \(45^\circ\) and \(135^\circ\) within \([0^\circ, 360^\circ)\).
5. Set \(\frac{x}{2} = \frac{\pi}{4}\) and \(\frac{x}{2} = \frac{3\pi}{4}\) (or \(\frac{x}{2} = 45^\circ\) and \(\frac{x}{2} = 135^\circ\)), then solve for \(x\) (or \(\theta\)) by multiplying both sides by 2. Make sure to check that the solutions lie within the original intervals \([0, 2\pi)\) for \(x\) and \([0^\circ, 360^\circ)\) for \(\theta\).

Key Concepts

Solving Trigonometric Equations

Solving trigonometric equations involves isolating the trigonometric function and finding all angle values within a specified interval that satisfy the equation. This often requires algebraic manipulation and using known values or identities to simplify the equation.

Properties of the Sine Function

The sine function is periodic with period 2π and has a range of [-1, 1]. Understanding its symmetry and key values at standard angles helps in finding exact solutions. For example, sin(θ) = sin(π - θ) and sin(θ) = sin(θ + 2πk) for any integer k.

Interval Notation and Angle Measurement

The problem specifies solutions over intervals [0, 2π) for radians and [0°, 360°) for degrees. Knowing how to convert between radians and degrees and restricting solutions to these intervals ensures all valid solutions are found without repetition.