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

Verified step by step guidance

1. Step 1: Recall the definitions of sine and cosecant. The sine function, \( \sin \theta \), is defined as the ratio of the opposite side to the hypotenuse in a right triangle. The cosecant function, \( \csc \theta \), is the reciprocal of sine, so \( \csc \theta = \frac{1}{\sin \theta} \).
2. Step 2: Given \( \sin \theta = \frac{1}{2} \), calculate \( \csc \theta \) using the reciprocal relationship: \( \csc \theta = \frac{1}{\sin \theta} = \frac{1}{\frac{1}{2}} = 2 \).
3. Step 3: Compare the calculated value of \( \csc \theta \) with the given value. Since both are equal to 2, the statement is consistent.
4. Step 4: Consider the possible angles \( \theta \) where \( \sin \theta = \frac{1}{2} \). These angles are typically \( \theta = 30^\circ \) or \( \theta = 150^\circ \) in the unit circle.
5. Step 5: Conclude that the statement \( \sin \theta = \frac{1}{2} \) and \( \csc \theta = 2 \) is possible, as both conditions are satisfied for the same angle \( \theta \).

Key Concepts

Sine and Cosecant Functions

The sine function, sin θ, represents the ratio of the length of the opposite side to the hypotenuse in a right triangle. The cosecant function, csc θ, is the reciprocal of sine, defined as csc θ = 1/sin θ. Understanding these relationships is crucial for determining the validity of the given statements.

Range of Trigonometric Functions

The sine function has a range of values between -1 and 1, meaning sin θ = 1/2 is a valid statement since 1/2 falls within this range. Conversely, csc θ, being the reciprocal of sine, has a range of values outside the interval [-1, 1]. This means that if sin θ = 1/2, then csc θ must equal 2, which is valid.

Trigonometric Identities

Trigonometric identities are equations that involve trigonometric functions and are true for all values of the variables involved. The identity csc θ = 1/sin θ is fundamental in verifying the relationship between sine and cosecant. Recognizing these identities helps in confirming whether the statements about sin θ and csc θ are consistent.