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

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Recall the definition of the cosecant function: \(\csc \theta = \frac{1}{\sin \theta}\).

Since \(\csc \theta = 100\), this means \(\sin \theta = \frac{1}{100} = 0.01\).

Consider the range of the sine function: \(-1 \leq \sin \theta \leq 1\). Since \(0.01\) lies within this range, it is a valid sine value.

Therefore, it is possible for \(\csc \theta\) to equal 100 because \(\sin \theta\) can be \(0.01\).

To find the specific angle(s) \(\theta\), you would use the inverse sine function: \(\theta = \sin^{-1}(0.01)\), keeping in mind the periodicity and symmetry of the sine function.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Definition and Range of Cosecant Function

The cosecant function, csc θ, is the reciprocal of the sine function, defined as csc θ = 1/sin θ. Since sine values range between -1 and 1, the cosecant values must be either greater than or equal to 1 or less than or equal to -1, never between -1 and 1.

Possible Values of Trigonometric Functions

Trigonometric functions have specific ranges that determine which values are possible. For csc θ, values like 100 are possible because they are greater than 1, meaning there exists an angle θ where sin θ = 1/100, which is within the sine function's range.

Reciprocal Relationship Between Sine and Cosecant

Since csc θ = 1/sin θ, understanding the reciprocal relationship helps in determining the feasibility of a given csc value. If csc θ = 100, then sin θ = 1/100, a small but valid sine value, confirming the statement is possible.