Problem 13 - Algebra and Trigonometry with Analytic Geometry

Question

Find the exact value. (a) \(\cot 120^{\circ}\) (b) \(\cot \left(-150^{\circ}\right)\)

Step-by-Step Solution

Answer

(a) \(-\frac{\sqrt{3}}{3}\); (b) \(\sqrt{3}\).

Step 1: Understanding the Cotangent Function

The cotangent of an angle \( \theta \) is the reciprocal of the tangent function, or \( \cot \theta = \frac{1}{\tan \theta} \). It can also be defined in terms of the sine and cosine functions: \( \cot \theta = \frac{\cos \theta}{\sin \theta} \). We will use these definitions to find the values for the given angles.

Step 2: Find Cotangent for \(120^{\circ}\)

\(120^{\circ}\) is in the second quadrant where sine is positive and cosine is negative. The reference angle for \(120^{\circ}\) is \(60^{\circ}\). Thus, \( \sin 120^{\circ} = \sin 60^{\circ} = \frac{\sqrt{3}}{2} \) and \( \cos 120^{\circ} = -\cos 60^{\circ} = -\frac{1}{2} \). Hence, \( \cot 120^{\circ} = \frac{\cos 120^{\circ}}{\sin 120^{\circ}} = \frac{-\frac{1}{2}}{\frac{\sqrt{3}}{2}} = -\frac{1}{\sqrt{3}} = -\frac{\sqrt{3}}{3} \).

Step 3: Find Cotangent for \(-150^{\circ}\)

\(-150^{\circ}\) is in the third quadrant when considering a positive angle equivalent, specifically \(210^{\circ}\). In the third quadrant, both sine and cosine are negative. The reference angle is \(30^{\circ}\), so \( \sin 210^{\circ} = -\sin 30^{\circ} = -\frac{1}{2} \) and \( \cos 210^{\circ} = -\cos 30^{\circ} = -\frac{\sqrt{3}}{2} \). Thus, \( \cot (-150^{\circ}) = \frac{\cos 210^{\circ}}{\sin 210^{\circ}} = \frac{-\frac{\sqrt{3}}{2}}{-\frac{1}{2}} = \sqrt{3} \).

Key Concepts

Cotangent Function

In trigonometry, the cotangent function is a fundamental trigonometric ratio. It is defined as the reciprocal of the tangent function. This means that if you know the tangent of an angle, you can easily find the cotangent by taking the reciprocal, as expressed by the formula:

• \( \cot \theta = \frac{1}{\tan \theta} \)

Moreover, the cotangent can also be expressed in terms of sine and cosine, which are the primary trigonometric functions:

• \( \cot \theta = \frac{\cos \theta}{\sin \theta} \)

This definition is particularly useful when we know the values of sine and cosine for a given angle, allowing us to quickly calculate the cotangent. For example, in the case of \(120^{\circ}\) or \(-150^{\circ}\), knowing the cosine and sine will directly lead us to finding the exact value of the cotangent for these angles.Using these definitions makes understanding and calculating the cotangent of angles more manageable. It sets the basis for exploring more complex concepts in trigonometry, especially when combined with understanding reference angles and reciprocal trigonometric functions.

Reference Angles

Reference angles are crucial in trigonometry to aid in evaluating trigonometric functions such as cotangent. A reference angle is defined as the acute angle that an angle makes with the x-axis. It is always measured from the x-axis to the terminal side of the angle, and it is always between \(0^{\circ}\) and \(90^{\circ}\).For any given angle, finding the reference angle helps you work with angles that are positioned in any of the four quadrants on the unit circle.

• For example, the reference angle for \(120^{\circ}\) is \(60^{\circ}\), because \(120^{\circ}\) is \(180^{\circ}\) minus \(60^{\circ}\).
• Similarly, the reference angle for \(-150^{\circ}\), which is equivalent to \(210^{\circ}\) in a full circle, is \(30^{\circ}\).

Reference angles simplify computations and understanding of trigonometric functions, as they allow us to use known values and identities to easily calculate functions of larger or negative angles.

Reciprocal Trigonometric Functions

Reciprocal trigonometric functions provide a valuable way of viewing and calculating trigonometric relationships. These functions include cotangent, secant, and cosecant, which are the reciprocals of tangent, cosine, and sine, respectively.

• Remember from earlier, the cotangent function \( \cot \theta = \frac{1}{\tan \theta} \), simplifies calculations whenever the tangent is known.
• The secant function is defined as \( \sec \theta = \frac{1}{\cos \theta} \).
• The cosecant function is expressed as \( \csc \theta = \frac{1}{\sin \theta} \).

These reciprocal functions allow for a complete understanding of trigonometric identities and serve as essential tools when working across different trigonometric equations and problems.Understanding reciprocal functions is especially useful when a function's value might be more accessible or suitable for given conditions, like multiples of known angle values and their reciprocal relationships in complex mathematical and scientific applications.