Problem 31 - Algebra 2 and Trigonometry

Question

In \(3-38,\) find each function value to four decimal places. $$ \cot 63^{\circ} $$

Step-by-Step Solution

Answer

The value of \( \cot 63^{\circ} \) is approximately 0.5095.

Step 1: Understanding the Function

The exercise requires us to find the value of the cotangent function, which is the reciprocal of the tangent function. That is, \( \cot \theta = \frac{1}{\tan \theta} \). In this case, we are seeking \( \cot 63^{\circ} \).

Step 2: Calculate \( \tan 63^{\circ} \) Using a Calculator

Use a calculator to determine the value of \( \tan 63^{\circ} \). Ensure that your calculator is set to degree mode. You should find that \( \tan 63^{\circ} \approx 1.9626 \).

Step 3: Find the Reciprocal of \( \tan 63^{\circ} \)

To find \( \cot 63^{\circ} \), take the reciprocal of \( \tan 63^{\circ} \). Thus, \( \cot 63^{\circ} = \frac{1}{\tan 63^{\circ}} \). Substitute the found value of \( \tan 63^{\circ} \) into this expression: \( \cot 63^{\circ} = \frac{1}{1.9626} \approx 0.5095 \).

Step 4: Round to Four Decimal Places

The value of \( \cot 63^{\circ} \) calculated in the previous step is already rounded to four decimal places, so no further rounding is needed.

Key Concepts

Cotangent Function

The cotangent function is a fundamental concept in trigonometry, often represented as \( \cot \theta \). This function is defined as the reciprocal of the tangent function. Mathematically, this is expressed as \[\cot \theta = \frac{1}{\tan \theta}.\]

This definition means that the cotangent of an angle \( \theta \) can be calculated if you know the tangent of that angle. In specific scenarios, such as the exercise you are studying, understanding this relationship allows you to solve for one trigonometric function value given another.

In practical terms:

• If you have the tangent of an angle, simply take its reciprocal to find the cotangent.
• It is crucial to ensure that your calculator is in the correct mode, typically degree mode when working with degrees.

This approach highlights the importance of reciprocal functions in trigonometry, making them highly useful for solving various problems.

Tangent Function

The tangent function, denoted as \( \tan \theta \), is one of the primary trigonometric functions. It plays a crucial role in understanding angles and their relationships in trigonometry.

The tangent of an angle \( \theta \) can be thought of as the ratio of the opposite side to the adjacent side in a right-angled triangle. Formally, this can be stated as:

• \( \tan \theta = \frac{\text{Opposite side}}{\text{Adjacent side}} \)
• Alternatively, in the unit circle context, \( \tan \theta \) is the ratio of the \( y \)-coordinate to the \( x \)-coordinate of a point.

This understanding of \( \tan \theta \) makes it easier to connect the tangent with real-world measurements and calculations. Knowing \( \tan 63^{\circ} \) is 1.9626, for example, provides a foundational basis for finding other related trigonometric values.

The power of the tangent function is its ability to convey both directional changes and heights within triangles and other geometric shapes.

Reciprocal Identities in Trigonometry

Reciprocal identities are a key concept in trigonometry. They allow mathematicians and students to switch between trig functions easily. These identities are vital tools as they simplify complex problem-solving processes.

In trigonometry, each main trigonometric function and its reciprocal function are paired:

• The secant function is the reciprocal of the cosine function: \( \sec \theta = \frac{1}{\cos \theta} \)
• The cosecant function is the reciprocal of the sine function: \( \csc \theta = \frac{1}{\sin \theta} \)
• The cotangent function is the reciprocal of the tangent function: \( \cot \theta = \frac{1}{\tan \theta} \)

Understanding these identities helps solve equations and understand relationships between trigonometric functions more efficiently. For instance, the exercise you are working on involves using the definition of the cotangent as a reciprocal of the tangent, highlighting the practical application of these identities.

Leveraging reciprocal identities simplifies calculations and helps ensure accuracy when working through trigonometric problems.