Question

What is the range of the cosecant function?

Short answer

The range of the cosecant function is \((-∞, -1] ∪ [1, ∞)\).

Step-by-step solution

Understanding the Cosecant Function

The cosecant function is defined as the reciprocal of the sine function: \(\text{csc}(x) = \frac{1}{\text{sin}(x)}\). The sine function, \(\text{sin}(x)\), oscillates between -1 and 1 for all \(-\frac{\text{π}}{2} \leq x \leq \frac{\text{π}}{2}\).

Identifying Undefined Points

The cosecant function is undefined where the sine function is zero because division by zero is undefined. This occurs at \(x = kπ\) for integer values of \(k\). Hence, \(\text{sin}(x) = 0\) at these points, and \(\text{csc}(x)\) is not defined.

Finding the Range of Sine

The range of \( \text{sin}(x) \) is between -1 and 1, i.e., \(-1 \leq \text{sin}(x) \leq 1\). The sine function never actually reaches any value outside this interval.

Determining the Range of Cosecant

Since \(\text{csc}(x) = \frac{1}{\text{sin}(x)}\), as \( \text{sin}(x) \) approaches zero from the positive and negative sides, \(\text{csc}(x)\) approaches infinity and negative infinity, respectively. The cosecant function is therefore only defined for values where \( \text{sin}(x) \eq 0 \). Hence, the range of \( \text{csc}(x) \) are all real numbers except in the interval \((-1, 1)\), that is, \(\text{csc}(x) \in (-\text{∞}, -1] \cup [1, \text{∞})\).

Key concepts

trigonometric functions

Trigonometric functions are fundamental in mathematics and are commonly used to study relationships in triangles and model periodic phenomena such as sound waves. One of the basic trigonometric functions is the sine function, denoted as \(\text{sin}(x)\). This function produces values that range from -1 to 1 as \(x\) varies over its domain. Understanding and analyzing these functions help in solving various mathematical problems related to angles and oscillations. Trigonometric functions such as sine, cosine, and tangent form the basis for more complex functions like secant, cosecant, and cotangent.

reciprocal functions

Reciprocal functions are derived from the basic trigonometric functions. The cosecant function, for example, is the reciprocal of the sine function and is represented as \(\text{csc}(x) = \frac{1}{\text{sin}(x)}\). This means that wherever the sine function is non-zero, the cosecant function will mirror this relationship inversely. If \(\text{sin}(x)\) is small, \(\text{csc}(x)\) will be large, and vice versa. This reciprocal relationship means the behavior of \(\text{csc}(x)\) is dependent on the values of \(\text{sin}(x)\), particularly in how \(\text{csc}(x)\) becomes undefined where \(\text{sin}(x) = 0\).

undefined points

When a function has undefined points, it means there are certain values in the domain where the function does not produce a valid output. For the cosecant function \(\text{csc}(x)\), undefined points occur where the sine function is zero because division by zero is not possible. Specifically, \(\text{sin}(x) = 0\) at integer multiples of \(π\) (like \(x = kπ\) for any integer \(k\)). At these points, \(\text{csc}(x)\) becomes undefined. Recognizing these points is crucial for correctly plotting or interpreting the behavior of trigonometric and reciprocal functions.