In Exercises 55–58, use a graph to solve each equation for -2π ≤ x ≤ 2π. csc x = 1

Verified step by step guidance

1. Recall that the cosecant function is the reciprocal of the sine function, so \(\csc x = \frac{1}{\sin x}\). Therefore, the equation \(\csc x = 1\) can be rewritten as \(\frac{1}{\sin x} = 1\).
2. From the equation \(\frac{1}{\sin x} = 1\), multiply both sides by \(\sin x\) (noting \(\sin x \neq 0\)) to get \(1 = \sin x\).
3. Now, solve the equation \(\sin x = 1\) for \(x\) in the interval \(-2\pi \leq x \leq 2\pi\) by identifying where the sine function reaches the value 1 on its graph.
4. Recall that \(\sin x = 1\) at \(x = \frac{\pi}{2} + 2k\pi\) for any integer \(k\). Find all such \(x\) values within the given interval by substituting integer values for \(k\).
5. List all solutions found in the interval \(-2\pi \leq x \leq 2\pi\) as the final answer to the equation \(\csc x = 1\).

Key Concepts

Understanding the Cosecant Function

The cosecant function, csc x, is the reciprocal of the sine function, defined as csc x = 1/sin x. It is undefined where sin x = 0, and its values correspond to the reciprocal of sine values. Recognizing this relationship helps in solving equations involving csc x.

Graphing Trigonometric Functions

Graphing csc x involves plotting the reciprocal of the sine curve, which has vertical asymptotes where sine is zero. Understanding the shape and key points of the csc x graph allows one to visually identify solutions to equations like csc x = 1 within a given interval.

Solving Trigonometric Equations on a Given Interval

Solving csc x = 1 over -2π ≤ x ≤ 2π requires finding all x-values where csc x equals 1 within this domain. This involves identifying corresponding sine values (sin x = 1) and considering the periodicity of the sine and cosecant functions to list all valid solutions.