CONCEPT PREVIEW Determine whether each statement is possible or impossible. sin θ = 1/2 , csc θ = 2

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Recall the definitions of sine and cosecant functions: \(\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}\) and \(\csc \theta = \frac{1}{\sin \theta}\).

Given \(\sin \theta = \frac{1}{2}\), use the reciprocal identity to find \(\csc \theta\): \(\csc \theta = \frac{1}{\sin \theta} = \frac{1}{\frac{1}{2}}\).

Simplify the expression for \(\csc \theta\) to check if it equals 2, which is the value given in the problem.

Compare the calculated value of \(\csc \theta\) with the given value to determine if the statement is possible or impossible.

Conclude that if the values match, the statement is possible; if not, it is impossible.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Definition of Sine and Cosecant Functions

Sine (sin θ) is the ratio of the length of the opposite side to the hypotenuse in a right triangle. Cosecant (csc θ) is the reciprocal of sine, defined as 1/sin θ. Understanding this reciprocal relationship is essential to verify if given values for sin θ and csc θ are consistent.

Reciprocal Identities in Trigonometry

Reciprocal identities state that csc θ = 1/sin θ. This means if sin θ = 1/2, then csc θ must be 2. Recognizing these identities helps determine whether the given pair of values can coexist or if they contradict each other.

Range and Possible Values of Trigonometric Functions

The sine function ranges between -1 and 1, so sin θ = 1/2 is possible. Since csc θ is the reciprocal, its values are outside the interval (-1, 1), so csc θ = 2 is also possible. Understanding these ranges helps assess the feasibility of the given values.

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