Use the given information to find each of the following.
cos x/2 , given cot x = -3, with π/2 < x < π

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Identify the given information: \( \cot x = -3 \) and \( \frac{\pi}{2} < x < \pi \). This means \( x \) is in the second quadrant where sine is positive and cosine is negative.

Recall the identity relating cotangent to sine and cosine: \( \cot x = \frac{\cos x}{\sin x} \). Since \( \cot x = -3 \), we can write \( \frac{\cos x}{\sin x} = -3 \).

Express \( \cos x \) in terms of \( \sin x \): \( \cos x = -3 \sin x \). Use the Pythagorean identity \( \sin^2 x + \cos^2 x = 1 \) to find \( \sin x \) and \( \cos x \). Substitute \( \cos x = -3 \sin x \) into the identity to get \( \sin^2 x + (-3 \sin x)^2 = 1 \).

Solve for \( \sin x \) from the equation \( \sin^2 x + 9 \sin^2 x = 1 \), which simplifies to \( 10 \sin^2 x = 1 \). Then find \( \sin x \) considering the quadrant (second quadrant means \( \sin x > 0 \)).

Use the half-angle formula for cosine: \[ \cos \frac{x}{2} = \pm \sqrt{\frac{1 + \cos x}{2}} \]. Determine the correct sign of \( \cos \frac{x}{2} \) based on the quadrant where \( \frac{x}{2} \) lies (since \( \frac{\pi}{4} < \frac{x}{2} < \frac{\pi}{2} \), \( \cos \frac{x}{2} > 0 \)). Substitute the value of \( \cos x \) found earlier into this formula.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Cotangent and Its Relationship to Sine and Cosine

Cotangent (cot x) is the ratio of cosine to sine, cot x = cos x / sin x. Knowing cot x helps determine the values of sine and cosine by expressing one in terms of the other, which is essential for solving trigonometric expressions involving half-angles.

Quadrant and Sign Determination

The interval π/2 < x < π places angle x in the second quadrant, where sine is positive and cosine is negative. Understanding the quadrant is crucial for assigning correct signs to trigonometric values when calculating half-angle expressions.

Half-Angle Formulas for Cosine

The half-angle formula for cosine is cos(x/2) = ±√[(1 + cos x)/2]. The sign depends on the quadrant of x/2. Applying this formula requires first finding cos x, then determining the correct sign based on the angle's quadrant.

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