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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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.

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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.

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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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