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.

Verified step by step guidance

1. Recall that \( \csc \theta = \frac{1}{\sin \theta} \). Given \( \csc \theta = -1.45 \), solve for \( \sin \theta \) by taking the reciprocal: \( \sin \theta = \frac{1}{-1.45} \).
2. Since \( \theta \) is in quadrant III, both sine and cosine are negative. Use the Pythagorean identity \( \sin^2 \theta + \cos^2 \theta = 1 \) to find \( \cos \theta \). Substitute \( \sin \theta \) into the identity and solve for \( \cos \theta \).
3. Calculate \( \cos \theta \) using the identity: \( \cos \theta = -\sqrt{1 - \sin^2 \theta} \). Remember to take the negative square root because \( \cos \theta \) is negative in quadrant III.
4. Recall that \( \cot \theta = \frac{\cos \theta}{\sin \theta} \). Substitute the values of \( \cos \theta \) and \( \sin \theta \) into this formula.
5. Rationalize the denominator if necessary by multiplying the numerator and the denominator by the conjugate of the denominator.

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.