Problem 55 - 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) \(\csc 32^{\circ} 40^{\prime} 3^{\prime \prime}\) (b) \(\tan 44^{\circ} 28^{\prime} 16^{\prime \prime}\)

Step-by-Step Solution

Answer

The rounded values obtained are likely to be: \(\csc 32^{\circ} 40^{\prime} 3^{\prime \prime} = \csc(32.6675) \approx 1.8633\), \(\tan 44^{\circ} 28^{\prime} 16^{\prime \prime} = \tan(44.4711) \approx 0.9756\).

Step 1: Convert DMS to Decimal Degrees

First, convert the angles from degrees, minutes and seconds to decimal degrees. For \(\csc 32^{\circ} 40^{\prime} 3^{\prime \prime}\), the conversion will be as follows: \(32 + \frac{40}{60} + \frac{3}{3600} = 32.6675\) degrees. For \(\tan 44^{\circ} 28^{\prime} 16^{\prime \prime}\), the conversion will be as follows: \(44 + \frac{28}{60} + \frac{16}{3600} = 44.4711\) degrees.

Step 2: Compute Trigonometric Functions

Next, use a calculator to compute the \(\csc\) and \(\tan\) of the decimal degree equivalents obtained above. Remember to set your calculator to the correct angle mode. \(\csc(32.6675)\) and \(\tan(44.4711)\) are the values to compute.

Step 3: Round to Four Decimal Places

The final step is to round the results obtained in step 2 to four decimal places. This could depend on the specific calculator used, but typically the rounding function can be accessed through the 'Math' menu on most calculators.

Key Concepts

Converting DMS to Decimal Degrees

Understanding how to convert degrees, minutes, and seconds (DMS) to decimal degrees is essential when working with trigonometric functions. Degrees, minutes, and seconds are a form of angular measurement, where 1 degree is equal to 60 minutes and 1 minute is equal to 60 seconds. To convert an angle given in DMS to decimal degrees, follow this formula:
\[\text{Decimal Degrees} = \text{Degrees} + \frac{\text{Minutes}}{60} + \frac{\text{Seconds}}{3600}\].
For instance, if you have an angle of 32 degrees, 40 minutes, and 3 seconds, the conversion to decimal degrees would be calculated as \(32 + \frac{40}{60} + \frac{3}{3600} = 32.6675\) degrees. This step is crucial for using a scientific calculator because these devices typically require angle measurements to be in decimal form. To facilitate your understanding, here is a clear process to perform the conversion:

• Divide the number of minutes by 60 and add the result to the number of degrees.
• Divide the number of seconds by 3600 and add the result to the previous sum.
• The final sum is the value in decimal degrees.

It's important not to skip this conversion step when solving trigonometric problems, as using the incorrect angle form can lead to inaccuracies in your final answer.

Evaluating Csc (Cosecant) Function

The cosecant (csc) is one of the six fundamental trigonometric functions. Importantly, it is the reciprocal of the sine function. The cosecant of an angle in a right-angled triangle is defined as the ratio of the length of the hypotenuse to the length of the opposite side. In formula terms:
\[\csc(\theta) = \frac{1}{\sin(\theta)}\].
This value is key for certain calculus concepts and geometry theorems. To evaluate the cosecant function using a calculator, you usually need to find the inverse of the sine of the angle, as most scientific calculators do not have a direct cosecant function button. Here's how to do it:

• Ensure your calculator is in the correct angle mode (degrees if working with degrees).
• Input the angle's decimal degree equivalent. For example, for \(\csc 32^\circ 40' 3''\), you first convert to decimal then use \(32.6675\) degrees.
• Compute the sine function of the angle.
• Finally, calculate the reciprocal of the sine value to get the cosecant.

Remember to round off your answer to the requested decimal place for precision.

Evaluating Tan (Tangent) Function

The tangent (tan) function, another important trigonometric function, is the ratio of the opposite to the adjacent side in a right-angled triangle. Calculators typically include the tangent function, making its evaluation more straightforward compared to cosecant.
Here’s how you evaluate the tangent function:

• Convert the DMS measurement to decimal degrees, as discussed in the previous section.
• Ensure your calculator is in the appropriate mode (degrees or radians, depending on your problem).
• Enter the decimal degree measurement into your calculator and press the tangent function (tan) button.
• The displayed result is the tangent of the angle. Remember to round to the specified number of decimal places, which is common practice in most mathematical computations for consistency and to avoid propagation of error.

With these steps, for instance, evaluating \(\tan 44^\circ 28' 16''\) requires computing \(\tan(44.4711)\) after converting to decimal degrees. It's pivotal to recognize rounding to four decimal places is often sufficient for most practical applications, providing a balance between precision and usability.