Problem 31 - Algebra and Trigonometry

Question

Find the exact value of the trigonometric function. $$ \cot \left(-\frac{\pi}{4}\right) $$

Step-by-Step Solution

Answer

The exact value is -1.

Step 1: Identify the cotangent definition

The cotangent function is defined as the reciprocal of the tangent function. That is, \(\cot(\theta) = \frac{1}{\tan(\theta)}\). We need to find \(\cot(-\frac{\pi}{4})\).

Step 2: Determine the tangent value

The angle \(-\frac{\pi}{4}\) is equivalent to \(-45^\circ\). We know that \(\tan(\frac{\pi}{4}) = 1\). Since tangent is an odd function, \(\tan(-\frac{\pi}{4}) = -\tan(\frac{\pi}{4}) = -1\).

Step 3: Calculate the cotangent value

Using the definition of cotangent, we find \(\cot(-\frac{\pi}{4}) = \frac{1}{\tan(-\frac{\pi}{4})} = \frac{1}{-1} = -1\).

Key Concepts

Cotangent

The cotangent function is a fundamental concept in trigonometry. It is often denoted as \( \cot(\theta) \) and is defined as the reciprocal of the tangent function. This means that

• \( \cot(\theta) = \frac{1}{\tan(\theta)} \).
• It can be thought of as the ratio of the cosine to the sine: \( \cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)} \).

Understanding cotangent can help solve various trigonometric problems, especially when dealing with angles in different quadrants or finding the exact values of trigonometric expressions. Also, remember that the cotangent function, like tangent, is periodic with a period of \( \pi \). Knowing these properties aids in navigating complex problems with ease.

Tangent

The tangent function, symbolized as \( \tan(\theta) \), is a key player in the family of trigonometric functions. The tangent of an angle \( \theta \) is defined as:

• \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \).

Tangent is unique among trigonometric functions because it is an odd function. That means \( \tan(-\theta) = -\tan(\theta) \). This property makes tangent useful when dealing with negative angles. Moreover, the tangent function has asymptotes where \( \cos(\theta) = 0 \), adding a layer of complexity. Still, understanding when these occur can simplify solving trigonometric equations. Remembering that the tangent of \( \pm \frac{\pi}{4} \) is \( \pm 1 \) is vital as these are common angles in problems.

Angle Conversion

Trigonometry often involves converting angles between different units. The two most common units are degrees and radians. Another pivotal concept is understanding the sign of an angle in different quadrants.

• To convert from radians to degrees, multiply by \( \frac{180}{\pi} \).
• To convert from degrees to radians, multiply by \( \frac{\pi}{180} \).

For example, the angle \(-\frac{\pi}{4}\) radians converts to \(-45^\circ\) in degrees. Negative angles move clockwise from the positive x-axis, determining the direction in which the angle is measured. This conversion and understanding of direction are crucial when evaluating trigonometric function values like cotangent or tangent.

Reciprocal Identities

Reciprocal identities form a foundational aspect of trigonometry. These identities relate the sine, cosine, and tangent functions to their respective reciprocals: cosecant, secant, and cotangent. Specifically,

• \( \csc(\theta) = \frac{1}{\sin(\theta)} \)
• \( \sec(\theta) = \frac{1}{\cos(\theta)} \)
• \( \cot(\theta) = \frac{1}{\tan(\theta)} \)

These identities allow for transforming expressions and solving equations. They also help in simplifying and finding the exact values of trigonometric expressions. Understanding and using reciprocal identities effectively is a crucial skill in mastering trigonometry, as it assists in approaching both basic and advanced problems methodically.