Problem 238 - Precalculus

Question

For the following exercises, use a graphing calculator to evaluate. $$ \cot 140^{\circ} $$

Step-by-Step Solution

Answer

\( \cot 140^{\circ} \approx -1.1918 \)

Step 1: Understand the Cotangent Function

Cotangent is the reciprocal of the tangent function. Therefore, \( \cot(\theta) = \frac{1}{\tan(\theta)} \). We need to evaluate \( \cot(140^{\circ}) \), which means finding the reciprocal of the tangent of \(140^{\circ}\).

Step 2: Use a Calculator to Find \( \tan(140^{\circ}) \)

Enter \(140^{\circ}\) into your calculator in degree mode and find \( \tan(140^{\circ}) \). The tangent of an angle in the second quadrant is negative, so you should expect a negative value. \( \tan(140^{\circ}) \approx -0.8391\).

Step 3: Calculate the Reciprocal to Find \( \cot(140^{\circ}) \)

Since \( \cot(\theta) = \frac{1}{\tan(\theta)} \), calculate the reciprocal of the result from Step 2. Thus, \( \cot(140^{\circ}) = \frac{1}{-0.8391} \approx -1.1918 \).

Step 4: Verification (Optional)

Double-check by ensuring your calculator is in the correct mode (degree mode) and re-evaluate \( \tan(140^{\circ}) \) and \( \cot(140^{\circ}) \) for accuracy.

Key Concepts

Cotangent Function

The cotangent function is an important trigonometric function closely related to the tangent function. While tangent directly measures the ratio of the opposite to the adjacent side of a right triangle, the cotangent does the reciprocal of that relationship. Therefore, if you know the tangent of an angle, finding the cotangent is as simple as inverting or finding the reciprocal of the tangent value. For any angle \( \theta \), the cotangent is defined as:

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

In practical terms, this means if we know \( \tan(140^{\circ}) \), then \( \cot(140^{\circ}) \) can be determined by taking the reciprocal of that tan value. Since angles in trigonometric functions can also relate to positions in different quadrants on the unit circle, it's crucial to remember that the tangent values in the second quadrant (such as 140 degrees) are negative. This, in turn, results in negative values for cotangent as well.

Reciprocal Functions

Reciprocal functions are fundamental in trigonometry, helping to solve complex problems by simplifying expressions. A reciprocal function is an inverse of another function. For the six trigonometric functions, each has its reciprocal:

• For sine \( \sin(\theta) \), the reciprocal is cosecant \( \csc(\theta) \)
• For cosine \( \cos(\theta) \), it is secant \( \sec(\theta) \)
• And for tangent \( \tan(\theta) \), the reciprocal is cotangent \( \cot(\theta) \)

To compute the reciprocal, simply switch the numerator and denominator of the function. This relationship particularly stands out when analyzing functions or equations that involve their standard forms. In a situation like evaluating \( \cot(140^{\circ}) \), recognizing it as \( \frac{1}{\tan(140^{\circ})} \) helps streamline the calculation as long as you accurately compute the tangent first. The simplicity of reciprocal functions provides ease in transformations and graphing, critical in higher-level mathematics.

Graphing Calculator Usage

Effectively using a graphing calculator can significantly enhance your understanding of trigonometric functions and their evaluations. To successfully evaluate \( \cot(140^{\circ}) \) using a graphing calculator, you should ensure it is set to degree mode since this reflects the angle measurement required here. To set the mode:

• Access the calculator settings or mode function.
• Switch from radian to degree if necessary.

Once the calculator is correctly set, enter \(140^{\circ}\) to find \( \tan(140^{\circ})\). Given the angle is in the second quadrant, the tangent should naturally resolve to a negative number. If \( \tan(140^{\circ}) \approx -0.8391 \), you then find the reciprocal by calculating \( \frac{1}{-0.8391} \), approximating \( \cot(140^{\circ}) \) to \( -1.1918 \). When in doubt, recalibrating and re-evaluating using consistent calculator settings can verify the accuracy of your results, ensuring understanding and reliability.