Determine whether each statement is possible or impossible. See Example 4. csc θ = 100

Verified step by step guidance

1. Understand that the cosecant function, \( \csc \theta \), is the reciprocal of the sine function, \( \sin \theta \).
2. Recall that the range of the sine function, \( \sin \theta \), is between -1 and 1, inclusive.
3. Since \( \csc \theta = \frac{1}{\sin \theta} \), the values of \( \csc \theta \) will be greater than or equal to 1 or less than or equal to -1.
4. Consider whether \( \csc \theta = 100 \) falls within the possible range of values for \( \csc \theta \).
5. Conclude whether the statement \( \csc \theta = 100 \) is possible or impossible based on the range of \( \csc \theta \).

Key Concepts

Cosecant Function

The cosecant function, denoted as csc(θ), is the reciprocal of the sine function. It is defined as csc(θ) = 1/sin(θ). This means that for csc(θ) to be defined, sin(θ) must not be zero, as division by zero is undefined. The range of the cosecant function is all real numbers except for the interval (-1, 1), meaning csc(θ) can take any value greater than or equal to 1 or less than or equal to -1.

Range of Cosecant

The range of the cosecant function is critical in determining the validity of statements involving csc(θ). Since csc(θ) can only take values outside the interval (-1, 1), any statement claiming csc(θ) equals a value within this range is impossible. Therefore, when evaluating csc(θ) = 100, it is essential to recognize that 100 is a valid value within the range of the cosecant function.

Trigonometric Identities

Trigonometric identities are equations that involve trigonometric functions and are true for all values of the variables involved. Understanding these identities, such as the Pythagorean identities, can help in analyzing and solving trigonometric equations. In this context, knowing that csc(θ) = 100 implies sin(θ) = 1/100 can assist in determining the possible angles θ that satisfy this equation, reinforcing the importance of identities in trigonometric analysis.