Problem 15 - Algebra 2

Question

Find the exact value of each expression. If the expression is undefined, write undefined. $$ \cot 0^{\circ} $$

Step-by-Step Solution

Answer

The expression \(\cot 0^{\circ}\) is undefined, since it results in a divide-by-zero situation.

Step 1: Define \(\cot \) function in terms of \(\tan \)

To start solving the exercise, it is crucial to define the cotangent function in terms of the tangent function. We can do this by using the trigonometric identity \(\cot \theta = \frac{1}{\tan \theta}\).

Step 2: Evaluate \(\tan 0^{\circ}\)

Now, using the properties of the tangent function, we need to evaluate \(\tan 0^{\circ}\). The tangent of 0 degrees, \(\tan 0^{\circ}\), is 0.

Step 3: Evaluate \(\cot 0^{\circ}\)

Finally, we substitute the obtained value for \(\tan 0^{\circ}\) into the relationship defined at step 1. The cotangent of 0 degrees, \(\cot 0^{\circ}\), is obtained as \(\frac{1}{\tan 0^{\circ}} = \frac{1}{0}\).

Key Concepts

Cotangent Function

The cotangent function is a trigonometric function that is less often encountered than its counterpart, the tangent function. Represented by \( \cot \theta \), it is defined as the reciprocal of the tangent function. Hence, it can be expressed as \( \cot \theta = \frac{1}{\tan \theta} \). What this means practically is that when you know the value of tangent, you can easily find the cotangent by taking the inverse.

However, there are specific conditions to keep in mind. If the tangent is zero, the cotangent becomes undefined due to division by zero. This typically occurs at specific angles in the unit circle, such as at 0 degrees.

• Key property: \( \cot \theta = \frac{\text{Adjacent}}{\text{Opposite}} \)
• Circular angles where \( \cot \theta \) is undefined: multiples of 180 degrees.

The cotangent function plays a crucial role in solving various trigonometric equations and in the analysis of periodic phenomena.

Tangent Function

The tangent function is one of the fundamental trigonometric functions, alongside sine and cosine. In a right triangle, the tangent of an angle \( \theta \) is the ratio of the length of the opposite side to the adjacent side. This relationship is key to understanding the function: \( \tan \theta = \frac{\text{Opposite}}{\text{Adjacent}} \).

On the unit circle, which is a vital tool in trigonometry, the tangent of an angle can be visualized as the slope of the line that crosses the circle.

• Zero angles such as 0 or 180 degrees have a tangent of 0.
• Non-zero values like 90 or 270 degrees result in an undefined tangent since the division would involve a zero denominator.

Recognizing points where the tangent is zero is essential in calculations, as it directly affects the cotangent function.

Trigonometric Identities

Trigonometric identities are equations that involve trigonometric functions and are true for every value of the appearing variables where both sides are defined. These identities are beneficial for verifying and simplifying trigonometric expressions.

The identity used most often in exercises involving cotangent and tangent is \( \cot \theta = \frac{1}{\tan \theta} \). This identity allows for easy conversion between cotangent and tangent, crucial when dealing with problems that require evaluation of these functions:

• This identity is used to solve equations by transforming a more complex expression into a simpler one.
• It is also used in calculus to find derivatives or integrals of trigonometric functions.

Understanding and applying these identities can greatly simplify the process of solving trigonometric problems, as well as in disciplines that use trigonometry such as engineering and physics.