Use identities to solve each of the following. Rationalize denominators when applicable. See Examples 5–7. Find cot θ , given that csc θ = ―1.45 and θ is in quadrant III.

Key Concepts

Cosecant and Cotangent Functions

Cosecant (csc) is the reciprocal of sine, defined as csc θ = 1/sin θ. Cotangent (cot) is the reciprocal of tangent, defined as cot θ = cos θ/sin θ. Understanding these relationships is crucial for solving trigonometric problems, especially when given one function and needing to find another.

Quadrants and Sign of Trigonometric Functions

The unit circle is divided into four quadrants, each affecting the signs of trigonometric functions. In quadrant III, both sine and cosine are negative, which means cosecant and cotangent will also be negative. Recognizing the quadrant helps determine the signs of the trigonometric values involved in the problem.

Rationalizing Denominators

Rationalizing the denominator involves eliminating any radical expressions from the denominator of a fraction. This is often done by multiplying the numerator and denominator by a suitable expression. In trigonometry, this technique can simplify expressions and make calculations clearer, especially when dealing with trigonometric identities.