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

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.