Inverses of csc, sec, and cot

Learning Objectives Define the inverse cosecant (arccsc), inverse secant (arcsec), and inverse cotangent (arccot) functions. State the restricted domains and ranges for y = arccsc(x), y = arcsec(x), and y = arccot(x). Evaluate expressions involving inverse reciprocal trigonometric functions for exact values. Convert expressions with arccsc, arcsec, and arccot into equivalent expressions with arcsin, arccos, and arctan. Solve simple trigonometric equations for a principal value using inverse reciprocal functions. Simplify composite functions involving trigonometric and inverse reciprocal trigonometric functions. You know how to find sin(π/6), but how do you find the angle whose cosecant is 2? 🤔 Let's learn how to work backwards with the reciprocal trig functions! This...

TermDefinitionExample Inverse Cosecant (arccsc)The inverse function of cosecant, written as arccsc(x) or csc⁻¹(x). It answers the question, 'What angle has a cosecant of x?'.arccsc(2) = π/6 because csc(π/6) = 2. The output angle must be in the range [-π/2, 0) U (0, π/2]. Inverse Secant (arcsec)The inverse function of secant, written as arcsec(x) or sec⁻¹(x). It answers the question, 'What angle has a secant of x?'.arcsec(√2) = π/4 because sec(π/4) = √2. The output angle must be in the range [0, π/2) U (π/2, π]. Inverse Cotangent (arccot)The inverse function of cotangent, written as arccot(x) or cot⁻¹(x). It answers the question, 'What angle has a cotangent of x?'.arccot(1) = π/4 because cot(π/4) = 1. The output angle must be in the range (0, π). Domain Restri...

Inverse Cosecant Identity arccsc(x) = arcsin(1/x), for |x| ≥ 1 To find the inverse cosecant of a number, find the inverse sine of its reciprocal. This is the most common method for evaluation. Inverse Secant Identity arcsec(x) = arccos(1/x), for |x| ≥ 1 To find the inverse secant of a number, find the inverse cosine of its reciprocal. This simplifies calculation significantly. Inverse Cotangent Identity arccot(x) = arctan(1/x) for x > 0 arccot(x) = arctan(1/x) + π for x < 0 To find the inverse cotangent, you can use the inverse tangent of the reciprocal. Be careful to add π for negative inputs to ensure the result is in the correct range of (0, π).