Use identities to solve each of the following. Rationalize denominators when applicable. See Examples 5–7. Find csc θ , given that cot θ = ―1/2 and θ is in quadrant IV.

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Recall the definitions of the trigonometric functions involved: \( \cot \theta = \frac{\cos \theta}{\sin \theta} \) and \( \csc \theta = \frac{1}{\sin \theta} \). Since \( \cot \theta = -\frac{1}{2} \), we can write \( \frac{\cos \theta}{\sin \theta} = -\frac{1}{2} \).

Express \( \cos \theta \) in terms of \( \sin \theta \) using the given cotangent: \( \cos \theta = -\frac{1}{2} \sin \theta \).

Use the Pythagorean identity \( \sin^2 \theta + \cos^2 \theta = 1 \) to substitute \( \cos \theta \) and solve for \( \sin \theta \). This gives: \( \sin^2 \theta + \left(-\frac{1}{2} \sin \theta\right)^2 = 1 \).

Simplify the equation to find \( \sin^2 \theta \), then take the square root to find \( \sin \theta \). Remember to consider the sign of \( \sin \theta \) based on the quadrant: since \( \theta \) is in quadrant IV, \( \sin \theta \) is negative.

Finally, find \( \csc \theta = \frac{1}{\sin \theta} \) and rationalize the denominator if necessary to express the answer in simplest form.

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Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values of the variable where both sides are defined. Key identities like cot²θ + 1 = csc²θ allow us to relate cotangent and cosecant, enabling the calculation of one function given the other.

Sign of Trigonometric Functions in Quadrants

The sign of trigonometric functions depends on the quadrant in which the angle lies. In quadrant IV, sine and cosecant are negative, while cosine and secant are positive. This information is crucial to determine the correct sign of the solution when using identities.

Rationalizing Denominators

Rationalizing denominators involves eliminating radicals or complex expressions from the denominator of a fraction. This process simplifies the expression and is often required for final answers in trigonometry problems to maintain standard form.