Problem 33 - Algebra and Trigonometry Real Mathematics, Real People

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) \(\sec 42^{\circ} 12^{\prime}\) (b) \(\csc 48^{\circ} 7^{\prime} 30^{\prime \prime}\)

Step-by-Step Solution

Answer

The secant of \(42^{\circ}12'\) is approximately equal to A (rounded to four decimal places), and the cosecant of \(48^{\circ}7'30''\) is approximately equal to B (rounded to four decimal places).

Step 1: Convert Minutes and Seconds to Decimals

Start by converting the minutes and seconds in the given angle measurements into decimal form. The conversion can be done using the formula: \(degrees + \frac{minutes}{60} + \frac{seconds}{3600}\).\n\n(a) For \(\sec 42^{\circ} 12^{\prime}\), the converted angle is \(42 + \frac{12}{60} = 42.2^{\circ}\)\n\n(b) For \(\csc 48^{\circ} 7^{\prime} 30^{\prime \prime}\), the converted angle is \(48 + \frac{7}{60} + \frac{30}{3600} = 48.125^{\circ}\)

Step 2: Calculate the Secant and Cosecant Values

Now, calculate the secant and cosecant values using a calculator.\n\n(a) \(\sec 42.2^{\circ}\) is just the reciprocal of \(\cos 42.2^{\circ}\). On a calculator, find the cosine of 42.2 degrees and then find its reciprocal; let's denote it as A.\n\n(b) \(\csc 48.125^{\circ}\) is the reciprocal of \(\sin 48.125^{\circ}\). On a calculator, find the sine of 48.125 degrees and then find its reciprocal; let's denote it as B.

Step 3: Round the Results

Finally, round the results A and B to four decimal places as requested by the exercise.

Key Concepts

Secant Function

The secant function, often denoted as \( \sec(\theta) \), is one of the key trigonometric functions not as commonly used as sine or cosine but important in many areas of mathematics and physics. It is defined as the reciprocal of the cosine function. So, \( \sec(\theta) = \frac{1}{\cos(\theta)} \). This means that when you have an angle \( \theta \), you find the cosine of that angle first and then calculate its reciprocal.

Using calculators can often simplify finding this value especially when the angle is given in degrees, as in our example with \( \sec 42^{\circ} 12^{\prime} \). After converting \( 42^{\circ} 12^{\prime} \) into a decimal form which is \( 42.2^{\circ} \), you compute the cosine of \( 42.2^{\circ} \) and then take its reciprocal to find the secant value. This process involves:

• Entering the degree mode on your calculator (important for getting the correct value).
• Finding \( \cos(42.2^{\circ}) \).
• Calculating \( \frac{1}{\cos(42.2^{\circ})} \) to obtain the value of \( \sec 42^{\circ} 12^{\prime} \).

Once calculated, make sure to round it to the desired number of decimal places, usually specified in problems.

Cosecant Function

The cosecant function, symbolized as \( \csc(\theta) \), is similar to the secant function but is the reciprocal of the sine function. It is defined as \( \csc(\theta) = \frac{1}{\sin(\theta)} \). This function, much like secant, is not included on many calculators directly, so the reciprocal needs to be calculated by first finding \( \sin(\theta) \).

In our exercise, the angle for the cosecant is \( 48^{\circ} 7^{\prime} 30^{\prime\prime} \). Converting this to a decimal form gives \( 48.125^{\circ} \). Here's how you find the cosecant of this angle using a calculator:

• Ensure your calculator is set to degree mode for accurate calculations.
• Calculate \( \sin(48.125^{\circ}) \).
• Take the reciprocal of \( \sin(48.125^{\circ}) \) to get \( \csc 48^{\circ} 7^{\prime} 30^{\prime\prime} \).

Remember, rounding to four decimal places helps in maintaining precision as required by most standard problems.

Angle Conversion

Converting angles from degrees, minutes, and seconds to a decimal form is an essential skill in trigonometry. It allows for simpler calculations and is crucial when using calculators that do not directly accept angles in minutes and seconds. The formula used for conversion is: \[\text{Decimal Degrees} = \text{Degrees} + \frac{\text{Minutes}}{60} + \frac{\text{Seconds}}{3600}\]This formula systematically converts each part of the angle into a decimal and adds them up.

For example, when converting \( 42^{\circ} 12^{\prime} \), the steps would be:

• Start with the degrees (42).
• Convert minutes to a decimal by \( \frac{12}{60} = 0.2 \).
• Add it all together to form \( 42.2^{\circ} \).

For angles with seconds, such as \( 48^{\circ} 7^{\prime} 30^{\prime\prime} \), include the seconds conversion:

• Convert seconds to a decimal, \( \frac{30}{3600} = 0.00833 \).
• Combine it: \( 48 + 0.1167 + 0.00833 = 48.125^{\circ} \).

By mastering angle conversion, you enhance your calculations' accuracy, especially when interacting with trigonometric functions like secant and cosecant.