Problem 77 - A Graphical Approach to Precalculus with Limits

Question

Use a calculator to find a decimal approximation for each value. Give as many digits as your calculator displays. $$\sec 183^{\circ} 48^{\prime}$$

Step-by-Step Solution

Answer

The decimal approximation of \(\sec 183^\circ 48'\) is approximately -1.0053.

Step 1: Understand the Angle

The angle given is in degrees and minutes: \(183^\circ 48'\). To work with this angle, we need to convert the minutes into degrees. Recall that 60 minutes is equal to 1 degree.

Step 2: Convert Minutes to Degrees

Convert the minutes into degrees by dividing by 60. This gives: \(48' = \frac{48}{60} = 0.8^\circ\).

Step 3: Calculate Total Degrees

Add the converted degrees to the given degrees to find the total angle in degrees: \(183^\circ + 0.8^\circ = 183.8^\circ\).

Step 4: Calculate the Cosine of the Angle

Use a calculator to find the cosine of \(183.8^\circ\). Ensure that your calculator is in degree mode, not radian mode, and input \(\cos(183.8^\circ)\).

Step 5: Calculate the Secant

Secant is the reciprocal of the cosine. Therefore, \(\sec(183.8^\circ) = \frac{1}{\cos(183.8^\circ)}\). Use your calculator to find this value and round according to your calculator's display.

Key Concepts

Decimal Approximation

Decimal approximation helps us convert and express mathematical results in decimal form. Calculators approximate values using decimals because most trigonometric values are irrational. This means they have non-repeating, non-terminating decimal expansions.
When we ask for a decimal approximation, we're essentially simplifying the infinite nature of certain numbers into a form that's practical for calculation purposes. Here's how we typically approach decimal approximation:

• We perform the calculation using the calculator, which automatically outputs the decimal form.
• The precision we choose can vary: we might display the result up to two, four, or more decimal places, depending on the context and calculator settings.

It's crucial to ensure the calculator is set correctly to degree mode for trigonometric functions when dealing with angles measured in degrees.

Angle Conversion

Angle conversion is the process of translating different angle measurements to a uniform system that makes them easier to handle. Angles are commonly measured in degrees and minutes, or entirely in degrees.
Degrees are typically divided into minutes, where 60 minutes is equal to 1 degree. To convert from minutes to degrees:

• Divide the number of minutes by 60.

For instance, 48 minutes would be \[48' = \frac{48}{60}= 0.8^\circ\]This step is essential, as it transforms the angle into a single unit that calculators can process efficiently.
Given an angle of \(183^\circ 48'\), it becomes \(183.8^\circ\) when the minutes are converted and combined with the degrees.

Secant Function

The secant function is one of the six fundamental trigonometric functions. It is the reciprocal of the cosine function, represented as:\[\sec(\theta) = \frac{1}{\cos(\theta)}\]
When finding the secant of an angle, you need to first calculate the cosine of that angle. Once you have the cosine, simply take its reciprocal to find the secant.
For example, with our angle \(183.8^\circ\), we first compute \(\cos(183.8^\circ)\). After finding this value, the next step is to calculate the secant by taking its reciprocal. Be mindful to ensure the calculator is in the correct mode (degree mode) when performing these operations.
Using the secant function is particularly useful in solving equations where cosine might be zero or near-zero, as it reconfigures the function without undefined results. Always rely on your calculator to ensure precision and accuracy when the numbers get too complex.