Problem 78 - 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. $$\cos 421^{\circ} 30^{\circ}$$

Step-by-Step Solution

Answer

\(\cos(421^{\circ}30^{\circ}) = \cos(61.3^{\circ}) \approx 0.4848096202\)

Step 1: Convert to Proper Degree

The angle given is incorrectly formatted in the prompt. Let's assume it is supposed to be a single angle written as \(421.3^{\circ}\), since \(421^{\circ}30^{\circ}\) is not a standard way to describe angles. Additionally, since trigonometry often considers angles within \([0^{\circ}, 360^{\circ})\), we need to reduce the angle by the full circle to get an equivalent angle: \(421.3^{\circ} - 360^{\circ} = 61.3^{\circ}\). Thus, the angle we need to evaluate \(\cos \) is \(61.3^{\circ}\).

Step 2: Use Calculator for Cosine

Input \(61.3^{\circ}\) into your calculator and press the cosine function button (\(\cos\)). Ensure that the calculator is in degree mode, not radian mode. This will give you the decimal approximation of \(\cos(61.3^{\circ})\).

Step 3: Read and Record the Decimal Approximation

Carefully read the display of your calculator. It likely shows several digits after the decimal point. Record all the digits as the final answer.

Key Concepts

Angle Conversion

When dealing with angles, it's important to ensure that they are in the correct format and within the correct range for trigonometric functions. In this scenario, the angle given, \(421^{\circ}30^{\circ}\), isn't standard. Instead, we consider the angle as \(421.3^{\circ}\).

Next, we must reduce angles to fall within the typical trigonometric cycle of \(0^{\circ}\) to \(360^{\circ}\). This means subtracting multiples of \(360^{\circ}\) from our given angle:\[421.3^{\circ} - 360^{\circ} = 61.3^{\circ}.\]

This converted angle, \(61.3^{\circ}\), is now suitable to use for trigonometric calculations.

Cosine Calculation

Calculating the cosine of an angle is straightforward with a calculator. This process determines how much of the angle's shadow is cast horizontally. For this calculation, make sure your calculator is set to degree mode (not radian or any other mode).

Here's how you do it:

• Enter the angle \(61.3^{\circ}\) into your calculator.
• Press the "\(\cos\)" button, which will give you the cosine of the angle.

Always double-check that the calculator's settings are in degrees, as this can significantly affect the result. By doing these steps, you've translated the angle into a value that represents its cosine, a critical function in trigonometry to express dimensions of triangles.

Decimal Approximation

Once you've found the cosine on your calculator, you'll see a decimal number displaying the cosine's value. Trigonometric functions like cosine rarely yield whole numbers, thus the decimal format is essential.

Follow these steps for the best results:

• Note the entire string of digits after the decimal point that your calculator shows.
• Decimal approximation is vital for precision. Use as many digits as are displayed for an accurate representation.
• This approximate value can be useful in various applications, particularly in solving trigonometry problems where exact precision isn't achievable.

Cosine values represent an exact ratio, but since calculators provide a decimal approximation, using all visible digits ensures your answer is as precise as possible, based on the tool's capabilities.