Determine whether each statement is possible or impossible. See Example 4. cot θ = ―6

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Recall the definition of cotangent in terms of sine and cosine: \(\cot \theta = \frac{\cos \theta}{\sin \theta}\).

Understand the range of \(\cot \theta\): Since \(\cot \theta\) is the ratio of cosine to sine, it can take any real value except where \(\sin \theta = 0\) (which would make the expression undefined).

Given \(\cot \theta = -6\), note that this is a real number, so it is possible as long as \(\sin \theta \neq 0\).

To confirm, consider that \(\cot \theta = -6\) means \(\frac{\cos \theta}{\sin \theta} = -6\), which implies \(\cos \theta = -6 \sin \theta\).

Since both sine and cosine values range between -1 and 1, and the ratio can be any real number, this equation can be satisfied for some angle \(\theta\), making the statement possible.

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주요 개념

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

Definition of Cotangent

Cotangent (cot θ) is the reciprocal of the tangent function, defined as cot θ = adjacent/opposite or cot θ = 1/tan θ. It represents the ratio of the x-coordinate to the y-coordinate on the unit circle for angle θ.

Range of Cotangent Function

The cotangent function can take any real value from negative to positive infinity, as it is undefined only where sine θ = 0. Therefore, cot θ = -6 is possible since cotangent values are not restricted to a specific interval.

Determining Possibility of Trigonometric Values

To determine if a trigonometric value is possible, consider the function's domain and range. Values outside the range or at points where the function is undefined are impossible, while values within the range are possible.