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

Verified step by step guidance

1. insert step 1> Start by understanding the given information: cot x = -3 and \( \frac{\pi}{2} < x < \pi \). This means x is in the second quadrant where cosine is negative and sine is positive.
2. insert step 2> Use the identity \( \cot x = \frac{\cos x}{\sin x} \) to express \( \cos x \) and \( \sin x \) in terms of each other. Since \( \cot x = -3 \), we have \( \frac{\cos x}{\sin x} = -3 \).
3. insert step 3> 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 solve for \( \sin x \).
4. insert step 4> Once \( \sin x \) is found, use it to find \( \cos x \) using \( \cos x = -3 \sin x \).
5. insert step 5> Use the half-angle identity for cosine: \( \cos \frac{x}{2} = \pm \sqrt{\frac{1 + \cos x}{2}} \). Determine the correct sign based on the quadrant where \( \frac{x}{2} \) lies.

Key Concepts

Cotangent Function

The cotangent function, denoted as cot(x), is the reciprocal of the tangent function, defined as cot(x) = cos(x)/sin(x). It is particularly useful in trigonometric identities and can help determine the signs of sine and cosine based on the quadrant in which the angle lies. In this case, cot(x) = -3 indicates that sine and cosine have opposite signs, which is essential for finding the values of cos(x) and sin(x).

Half-Angle Formulas

Half-angle formulas are trigonometric identities that express the sine and cosine of half an angle in terms of the sine and cosine of the original angle. For example, cos(x/2) can be calculated using the formula cos(x/2) = ±√((1 + cos(x))/2). These formulas are particularly useful when the angle is halved, allowing for the calculation of trigonometric values without directly measuring the angle.

Quadrants and Angle Ranges

Understanding the unit circle and the corresponding quadrants is crucial in trigonometry. The range π/2 < x < π indicates that angle x is in the second quadrant, where cosine values are negative and sine values are positive. This knowledge helps determine the signs of the trigonometric functions involved, which is vital for accurately calculating cos(x/2) and ensuring the correct application of the half-angle formulas.