Problem 55 - Precalculus : Building Concepts and Connections

Question

Use a scientific calculator to evaluate the giren trigonometric functions to four decimal places. $$\cot 215^{\circ}$$

Step-by-Step Solution

Answer

The result cannot be determined unless the calculations are carried out with a specific scientific calculator. Assuming all the steps are carried out correctly, the cotangent of 215 degrees will be rounded to a value with four decimal places.

Step 1: Convert degrees to radians

The cotangent function on most calculators works with radians, not degrees. So first, convert 215 degrees to radians. For this, use the degree to radians formula \(r = d \times \frac{\pi}{180}\) where 'r' represents radians, 'd' represents the degree measure to be converted, and \(\pi\) is the mathematical constant. Plugging in 215 for 'd', you get \(r = 215 \times \frac{\pi}{180}\).

Step 2: Calculate Cotangent

To calculate the cotangent of a value, take the reciprocal of its tangent. First, calculate tan(r) (where r is the result from step 1) using the calculator, then take the reciprocal.

Step 3: Rounding the result

To conform to the problem's requirement, the result should be rounded to 4 decimal places. This can be done using the calculator's rounding feature.

Key Concepts

Cotangent Function

The cotangent function, often written as \(\cot\), is one of the six fundamental trigonometric functions. It is intimately related to the tangent function but with a twist. As the name suggests, \(\cot\) is the reciprocal of the tangent function. This means that for any angle \(\theta\), the cotangent of \(\theta\) can be expressed as \(\cot(\theta) = \frac{1}{\tan(\theta)}\).

Understanding the cotangent function also requires a grasp of the right-angled triangle. In a right-angled triangle, the cotangent of an angle is the ratio of the adjacent side to the opposite side. However, when evaluating cotangent using a calculator, we typically refer to the unit circle, where the angle is measured from the positive x-axis, and the cotangent is the x-coordinate divided by the y-coordinate.

Since the cotangent function is cyclic with a period of \(\pi\) radians, it repeats its values every \(\pi\) radians. This peculiarity can affect calculations and interpretations of cotangent values, especially when dealing with negative angles or angles greater than \(180^\circ\).

Degrees to Radians Conversion

Angles can be measured in degrees or radians, and converting between these two units is a fundamental skill in trigonometry. The conversion hinges on the relationship that \(180^\circ\) is equivalent to \(\pi\) radians. From this relationship, the conversion formula is derived: \(r = d \times \frac{\pi}{180}\), where \(r\) represents the angle in radians and \(d\) the angle in degrees.

In the context of the exercise, to convert \(215^\circ\) to radians, we multiply 215 by \(\frac{\pi}{180}\). Understanding this conversion is crucial because trigonometric functions on scientific calculators typically require angle inputs in radians. Therefore, mastering degrees to radians conversion is not just a theoretical exercise; it's a practical necessity for correctly solving trigonometry problems.

Scientific Calculator Usage

A scientific calculator is an indispensable tool for solving trigonometric functions, including the cotangent. It is essential to familiarize oneself with the calculator's functions, which vary by model, but generally follow similar steps for trigonometric calculations.

Here's how you can typically use a scientific calculator for trigonometry:

• First, ensure the calculator is in the correct mode (usually radians for trig functions).
• Next, input the angle after performing any necessary conversion from degrees to radians.
• Finally, select the trigonometric function you wish to calculate and press equals.

For cotangent specifically, most calculators require calculating the tangent and then taking its reciprocal. It’s important to note that some advanced scientific calculators have a direct cotangent function button, streamlining the process.

Reciprocal of Tangent

As mentioned, the cotangent function is the reciprocal of the tangent function. This reciprocal nature means that if you know the tangent of an angle, you can easily find the cotangent by flipping the tangent's value (i.e., \(\cot(\theta) = \frac{1}{\tan(\theta)}\)). Let's consider what this looks like in computation.

If we calculate the tangent of an angle and get a certain value, we obtain the cotangent by taking one over that value. In practical terms, if your calculator doesn't have a cotangent button, you would:

• Calculate \(\tan(\theta)\)
• Press the reciprocal key (often labeled as \(1/x\) or \(x^{-1}\))

This gives you \(\cot(\theta)\), which should then be rounded as per the requirements of your problem, in this case to four decimal places.