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

Verified step by step guidance

1. Step 1: Recall that the cosecant function, \( \csc x \), is the reciprocal of the sine function, \( \sin x \). Therefore, \( \csc x = 1 \) implies \( \sin x = 1 \).
2. Step 2: Identify the values of \( x \) where \( \sin x = 1 \) within the interval \(-2\pi \leq x \leq 2\pi\).
3. Step 3: Recognize that \( \sin x = 1 \) at \( x = \frac{\pi}{2} + 2k\pi \), where \( k \) is an integer.
4. Step 4: Determine the specific values of \( x \) by substituting integers for \( k \) that keep \( x \) within the interval \(-2\pi \leq x \leq 2\pi\).
5. Step 5: Verify these values by checking the graph of \( \sin x \) to ensure that \( \sin x = 1 \) at these points.

Key Concepts

Cosecant Function

The cosecant function, denoted as csc(x), is the reciprocal of the sine function. It is defined as csc(x) = 1/sin(x). Understanding this relationship is crucial for solving equations involving csc(x), as it allows us to translate the equation into a sine function, which can then be analyzed graphically.

Graphing Trigonometric Functions

Graphing trigonometric functions involves plotting the values of the function over a specified interval. For csc(x), the graph consists of vertical asymptotes where sin(x) = 0 and the function's values where sin(x) is non-zero. This visual representation helps identify the solutions to equations like csc(x) = 1 by finding the intersections with the horizontal line y = 1.

Periodic Nature of Trigonometric Functions

Trigonometric functions are periodic, meaning they repeat their values in regular intervals. The cosecant function has a period of 2π, which implies that solutions to equations can be found within one period and then extended to other periods. This property is essential for determining all solutions within the given interval of -2π to 2π.