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

Verified step by step guidance

1. Start by rewriting the given equation: \(\sin\left(\frac{\theta}{2}\right) = \csc\left(\frac{\theta}{2}\right)\). Recall that \(\csc x = \frac{1}{\sin x}\), so substitute to get \(\sin\left(\frac{\theta}{2}\right) = \frac{1}{\sin\left(\frac{\theta}{2}\right)}\).
2. Multiply both sides of the equation by \(\sin\left(\frac{\theta}{2}\right)\) to eliminate the fraction, giving \(\sin^2\left(\frac{\theta}{2}\right) = 1\).
3. Recognize that \(\sin^2 x = 1\) implies \(\sin x = \pm 1\). So, set \(\sin\left(\frac{\theta}{2}\right) = 1\) and \(\sin\left(\frac{\theta}{2}\right) = -1\) separately to find all possible solutions.
4. Solve each equation for \(\frac{\theta}{2}\) over the interval corresponding to \(\theta \in [0^\circ, 360^\circ)\), which means \(\frac{\theta}{2} \in [0^\circ, 180^\circ)\). Use the unit circle values where sine equals \(1\) or \(-1\) within this interval.
5. Finally, multiply each solution for \(\frac{\theta}{2}\) by 2 to find the values of \(\theta\) in the interval \([0^\circ, 360^\circ)\). These will be the exact solutions to the original equation.

Key Concepts

Trigonometric Functions and Their Definitions

Understanding sine (sin) and cosecant (csc) functions is essential. Sine of an angle is the ratio of the opposite side to the hypotenuse in a right triangle, while cosecant is its reciprocal, defined as 1/sin. Recognizing this reciprocal relationship helps in transforming and solving the equation.

Solving Trigonometric Equations

Solving equations like sin(θ/2) = csc(θ/2) involves rewriting the equation using identities, simplifying, and finding all angle solutions within the given interval. This process often requires considering domain restrictions and checking for extraneous solutions.

Interval and Angle Measurement Conventions

The problem specifies solutions over [0, 2π) radians and [0°, 360°) degrees. Understanding how to convert between radians and degrees and how to interpret these intervals ensures that all valid solutions are found and expressed correctly within the specified domain.