Which trigonometric function is the reciprocal of sine?

Which trigonometric function is the reciprocal of sine?

• A. Sine
• B. Cosecant
• C. Tangent
• D. Cotangent

Answer: (B)

The reciprocal of sine is the cosecant. Sine is opposite over hypotenuse, so its reciprocal is hypotenuse over opposite, which is defined as cosecant: csc θ = 1/sin θ. This relationship also shows up on the unit circle, where sin θ = y and csc θ = 1/y, meaning csc θ is undefined when sin θ = 0 (where y = 0). For example, if sin θ = 1/2, then csc θ = 2. This matches the idea that csc is the reciprocal of sin, not related to tangent or cotangent.

The reciprocal of sine is the cosecant. Sine is opposite over hypotenuse, so its reciprocal is hypotenuse over opposite, which is defined as cosecant: csc θ = 1/sin θ. This relationship also shows up on the unit circle, where sin θ = y and csc θ = 1/y, meaning csc θ is undefined when sin θ = 0 (where y = 0). For example, if sin θ = 1/2, then csc θ = 2. This matches the idea that csc is the reciprocal of sin, not related to tangent or cotangent.