Problem 53 - Precalculus: A Concise Course

Question

Use a calculator to evaluate each function. Round your answers to four decimal places. (Be sure the calculator is in the correct angle mode.) (a) \(\cot 11^{\circ} 15^{\prime}\) (b) \(\tan 11^{\circ} 15^{\prime}\)

Step-by-Step Solution

Answer

The results will be dependent on the calculator and its settings but with correctly done calculations and rounding of the results to four decimal places, the accurate answers for both should be obtained.

Step 1: Convert to Decimal Degrees

Firstly, convert the given angles from degree-minute notation to decimal degree notation. We achieve this by dividing the minutes by 60 and adding the result to the degrees. Thus, \(11^{\circ} 15^{\prime}\) becomes \(11.25^{\circ}\).

Step 2: Evaluate Cotangent

Now, ensure your calculator is in the correct degree mode. Evaluate \(\cot 11.25^{\circ}\) by typing this into your calculator.

Step 3: Evaluate Tangent

Again, ensuring your calculator is in the correct degree mode, evaluate \(\tan 11.25^{\circ}\) by typing this into your calculator.

Step 4: Round Results

Depending on your calculator, the results for both cotangent and tangent will be displayed. Now, round these to four decimal places.

Key Concepts

Cotangent Function

The cotangent function is one of the six primary trigonometric functions. It relates to the ratio of the adjacent side to the opposite side in a right-angled triangle. In mathematical terms, the cotangent of an angle \(\theta\) is written as \(\cot(\theta)\). This can also be described as the reciprocal of the tangent function:

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

Cotangent is often used in various branches of mathematics, such as in solving triangles or analyzing periodic functions. When working with cotangent, it's crucial to ensure your calculator is set in the correct angle mode (degrees or radians). For example, when evaluating \(\cot(11.25^{\circ})\), your calculator needs to be in degree mode to ensure accurate results. Once the angle is input correctly, the calculator will provide the necessary output that can then be rounded to the desired precision.

Tangent Function

The tangent function is another essential trigonometric function. It is represented by \(\tan(\theta)\) and is defined as the ratio of the opposite side to the adjacent side in a right-angle triangle. This function helps in determining the slope or angle of a line.

• \(\tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}}\)

Like the cotangent, the tangent function is critical in trigonometry, calculus, and physics. To use this function with a calculator, ensure the angle measure is correct and the calculator is set to the right mode. For example, to find \(\tan(11.25^{\circ})\), enter the equivalent decimal degree in the correct angle mode. The calculator computes the value, which can then be refined to the required number of decimal places.

Angle Conversion

Before calculating trigonometric functions, angles often need to be converted to a suitable form. Degrees and minutes must be transformed into decimal degrees to simplify computations. Converting is straightforward:

• Convert minutes by dividing by 60 and adding to the degrees.

For instance, with the angle \(11^{\circ} 15^{\prime}\):

• Minutes are divided: \(15 \div 60 = 0.25\)
• Added to degrees: \(11 + 0.25 = 11.25^{\circ}\)

This conversion is vital to ensure calculator accuracy and facilitate easy computation of trigonometric values.

Calculator Use

Calculator use is crucial in evaluating trigonometric functions accurately. Modern calculators allow easy computation of functions like tangent and cotangent, provided they are set up correctly. Here are some important tips:

• Ensure that your calculator is in the correct angular mode (Degrees for this problem).
• Use the correct buttons for cotangent or compute its reciprocal using the tangent function.
• Input the angle in the decimal degree format (e.g., 11.25°).
• Make sure the calculator display is clear to avoid mistakes from prior calculations.

Using a calculator requires practice, but by following these guidelines, you will obtain precise answers promptly.

Rounding Numbers

Rounding numbers in mathematics is an important step, especially when dealing with trigonometric functions where precision is key. Often, results from a calculator must be rounded to a specific number of decimal places based on the problem's requirements.

• Round by looking at the digit one place beyond your target decimal spot.
• If this digit is 5 or more, add one to the target spot.
• If it's less than 5, keep the target number as it is.

For instance, if you get a result of 3.8976 and need to round to four decimal places, your result remains 3.8976. Understanding the rounding rules allows you to present your answers accurately, which is essential for achieving the correct solution in homework and exams.