Problem 99 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use a calculator to evaluate the trigonometric function. Round your answer to four decimal places. (Be sure the calculator is set in the correct angle mode.) $$\tan (2 \pi / 9)$$

Step-by-Step Solution

Answer

The value of \(\tan(2\pi/9)\), rounded to four decimal places, is 0.3639 (Note: Slight variations may occur depending on the calculator or software used for calculations.)

Step 1: Check Calculator Mode

Before beginning with the calculation, ensure that your calculator is set to radian mode as the input angle in the problem is given in radians.

Step 2: Input the Function

Now, input the function \(\tan(2\pi/9)\) into your calculator. Ensure to use parenthesis correctly to avoid any incorrect calculations.

Step 3: Calculate the Value and Round Off

After the input, calculate the value. You'll get an output from your calculator. Round off this output to four decimal places.

Key Concepts

Trigonometric Calculator Usage

Understanding how to use a trigonometric calculator is essential for tackling a variety of math and engineering problems. When solving for an unknown trigonometric function like the \(\tan (2 \pi / 9)\), the first step is to correctly input the function into your calculator. Ensure that you are familiar with the calculator's syntax for inputting trigonometric functions.

Start by identifying the trigonometric function key, often abbreviated as 'TAN.' Enter the angle next, making sure to use parentheses to denote the specific portion of the angle you are evaluating. For our problem, you would input \(\tan(\), followed by the angle in parentheses \((2\pi/9)\), and close off with another parenthesis. Confirm that each symbol and number is inputted correctly; this attention to detail is key to obtaining the correct solution.

Common Pitfalls to Avoid

• Incorrectly inputting the angle with the wrong unit or format.
• Not using parentheses can lead to improper order of operations, yielding incorrect results.

Once entered, the calculator will process the information and provide you with the trigonometric function's value.

Radian Mode

Trigonometric functions can be evaluated in different angle units, the most common being degrees and radians. Our example requires the use of radian mode. To prevent mistakes, check that your calculator is set to radian mode prior to calculations.

Radians and degrees are simply different ways of measuring angles: 360 degrees is equivalent to \(2\pi\) radians. Setting your calculator to the correct mode is crucial as it directly influences the output result. In the case of \(\tan(2\pi/9)\), the angle is inherently given in radians, so the calculator must be in radian mode to interpret the angle value as intended.

Switching to Radian Mode

Typically, this can be done by accessing the mode settings of the calculator and selecting 'RAD' or 'RADIANS'. Always double-check the settings to avoid taking trigonometric values in the wrong mode.

Rounding Decimal Places

Once you've calculated the trigonometric value, the last critical step is to round it to the specified number of decimal places, which ensures the preciseness of your answer. Rounding is a way to simplify a number, making it easier to work with, without losing much precision.

For the problem at hand, the answer needs to be rounded to four decimal places. This means you look at the fifth decimal place, and if it's 5 or above, you increase the fourth decimal place by one; if it's below 5, you leave the fourth decimal place as is. Here's a simplified guide to rounding:

Steps for Rounding

• Identify the decimal place to which you need to round.
• Look at the next decimal place for rounding guidance.
• If this number is 5 or more, round up the previous digit.
• If this number is less than 5, keep the previous digit the same.
• Discard all digits following the digit you are rounding to.

Apply this to the solution from your calculator, and the result will be the trigonometric function value, rounded to four decimal places.