Problem 40 - Algebra and Trigonometry

Question

Use a calculator to find the value of the trigonometric function to four decimal places. $$ \cos 21^{\circ} $$

Step-by-Step Solution

Answer

The value of \( \cos 21^{\circ} \) to four decimal places is 0.9336

Step 1: Set the calculator to degrees mode

Before starting, be sure that your calculator is in degrees mode since the angle is given in degrees. Most calculators have the option to switch between degrees and radians.

Step 2: Input the function

Now input '\(\cos 21^\circ\)' into the calculator

Step 3: Evaluate the function

After inputting the function, press the equal button to get the answer. Remember to round off to four decimal places.

Key Concepts

Calculator Usage

Using a calculator to find the value of trigonometric functions is a straightforward process, but it can vary slightly depending on your calculator model. The basic steps usually involve:

• Turning on the calculator and clearing any previous entries or settings.
• Ensuring the calculator is set to the proper mode for angles, such as degrees or radians.
• Pressing the keys in the correct order to enter the trigonometric function.
• Pressing the "equals" or "enter" button to perform the calculation.

Before starting any trigonometric calculations, a few practice runs can help ensure you’re comfortable with the specific button sequences and settings on your calculator. This will save time and reduce errors during more complex exercises.

Degrees Mode

When working with angles, it’s essential to understand the difference between degrees and radians. Angles can be measured in both, but if you are dealing with a problem provided in degrees, like \(21^\circ\), your calculator must be in degrees mode. Here’s how to check:

• Find the mode button on your calculator, which might be labeled as "MODE," "2nd," or a similar function.
• Look for options on the screen displaying "Deg" or "Rad." Select "Deg" if working in degrees.

It's important to regularly check your calculator's mode, especially if it's used by multiple people or if it retains settings from previous operations. This precaution prevents incorrect results and helps ensure you’re performing calculations based on the right units of measure.

Cosine Function

The cosine function is a fundamental trigonometric function, often written as \(\cos\theta\), where \(\theta\) is the angle in question. It’s part of the primary trigonometric ratios and deals with the ratio of the adjacent side to the hypotenuse in a right triangle.

To compute \(\cos 21^\circ\), you:

• Ensure your calculator is in degrees mode.
• Press the "COS" button, usually easily found on the calculator.
• Enter \(21\) and then close any necessary brackets if your calculator requires it.
• Press "equals" or "enter" to perform the calculation.

Understanding the roles of trigonometric functions like cosine helps in solving various mathematical and real-world problems, such as determining distances and angles in navigation or physics.

Rounding Off

Rounding is crucial when dealing with decimals in mathematical calculations, particularly when precision to a specific decimal place is required. You may need to round off numbers frequently after performing operations on a calculator.

In this exercise, you are instructed to round the result to four decimal places:

• Look at the fifth decimal place. If it's 5 or more, increase the fourth decimal place by 1.
• If the fifth decimal is less than 5, simply leave the fourth decimal as it is.

Applying correct rounding ensures that the answers remain accurate and within acceptable precision, especially important in scientific and engineering calculations where small errors could result in significant consequences.