Use the given information to find each of the following.
cot θ/2, given tan θ = -(√5)/2 , with 90° < θ < 180°

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Identify the given information: \(\tan \theta = -\frac{\sqrt{5}}{2}\) and the angle \(\theta\) is in the second quadrant, where \(90^\circ < \theta < 180^\circ\).

Recall the half-angle identity for cotangent: \(\cot \frac{\theta}{2} = \pm \sqrt{\frac{1 + \cos \theta}{1 - \cos \theta}}\). The sign depends on the quadrant of \(\frac{\theta}{2}\).

Determine the quadrant of \(\frac{\theta}{2}\). Since \(\theta\) is between \(90^\circ\) and \(180^\circ\), \(\frac{\theta}{2}\) is between \(45^\circ\) and \(90^\circ\), which is the first quadrant where cotangent is positive.

Find \(\cos \theta\) using the identity \(\tan^2 \theta + 1 = \frac{1}{\cos^2 \theta}\). Rearrange to solve for \(\cos \theta\) and use the sign of cosine in the second quadrant (which is negative).

Substitute the value of \(\cos \theta\) into the half-angle formula for \(\cot \frac{\theta}{2}\) and simplify to express \(\cot \frac{\theta}{2}\) in terms of known quantities.

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Key Concepts

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

Understanding the Given Angle and Its Quadrant

The angle θ lies between 90° and 180°, placing it in the second quadrant where sine is positive and cosine is negative. Knowing the quadrant helps determine the signs of trigonometric functions, which is essential when calculating values like cotangent of half the angle.

Half-Angle Identities in Trigonometry

Half-angle identities relate the trigonometric functions of θ/2 to those of θ. For cotangent, the identity cot(θ/2) = (1 + cos θ) / sin θ is commonly used. Applying these identities requires knowing or finding sine and cosine of θ first.

Converting Between Tangent, Sine, and Cosine

Given tan θ, sine and cosine can be found using the Pythagorean identity and the sign conventions of the quadrant. Since tan θ = sin θ / cos θ, knowing tan θ and the quadrant allows calculation of sin θ and cos θ, which are needed for half-angle formulas.

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