Problem 100 - 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.) $$\tan (11 \pi / 9)$$

Step-by-Step Solution

Answer

The value of \(\tan (\frac{11\pi}{9})\) rounded to four decimal places is -0.1324.

Step 1: Set Calculator Mode

Set your calculator to radian mode. This is because the problem gives the angle in radians. Many calculators are set to degree mode by default, so ensure to switch to radian mode for this problem.

Step 2: Evaluate Trigonometric Function

With the calculator in radian mode, input \(\tan (\frac{11\pi}{9})\). This will give the tangent of the given angle.

Step 3: Round Answer

Round the answer given by the calculator to four decimal places. This is because the problem specifically asked for the answer to be rounded in this way.

Key Concepts

Tangent Function

The tangent function, often denoted as \( \tan \theta \), is a fundamental concept in trigonometry. Understanding it is crucial for solving many mathematical problems involving angles. Essentially, the tangent of an angle in a right triangle is the ratio of the length of the opposite side to the adjacent side. This can be expressed as:

• \( \tan \theta = \frac{\text{opposite}}{\text{adjacent}} \)

However, when dealing with the unit circle, the tangent function can also be described using sine and cosine:

• \( \tan \theta = \frac{\sin \theta}{\cos \theta} \)

The tangent function is periodic, with a period of \( \pi \). This means that \( \tan(\theta) = \tan(\theta + n\pi) \) for any integer \( n \). As shown in the given problem with \( \tan(11\pi/9) \), finding the tangent requires using the function's properties and understanding that angles can be in radians.

Angle in Radians

Angles can be represented in either degrees or radians. Each mode of measuring angles is widely used in different contexts.
Radians are particularly common in higher mathematics and physics because they allow for more straightforward mathematical manipulation.
One full circle is \(2\pi\) radians, which equates to 360 degrees. Therefore, one radian is equal to \(\frac{180}{\pi}\) degrees.

• For example, \(\pi\) radians is equivalent to 180 degrees.
• \( \frac{\pi}{2} \) radians is equivalent to 90 degrees.

In the exercise, the angle \( 11\pi/9 \) is expressed in radians. This is a crucial point because calculators need to be set to the correct mode (radian or degree) to ensure valid results. Working with radians allows us to easily relate angles to lengths on the unit circle, making calculations more intuitive for continuous functions like trigonometry.

Calculator Usage

To evaluate trigonometric functions accurately, it is vital to correctly set up and use your calculator. Here's a simple guide:
Ensure your calculator is in the correct mode for the angle unit you are using. In most cases, as with the exercise provided, you will need to switch from degrees to radians:

• Most calculators default to degree mode, so check the mode setting before proceeding.
• Switching to radian mode is usually a simple button press, often labeled "mode" on the calculator.

Once in the correct mode, input the function as it is given. For \( \tan(11\pi/9) \):

• Input \( 11 \times \pi \div 9 \) first to handle the radian conversion internally.
• Follow it by applying the tangent function \( \tan() \) to get the result.

Finally, rounding the answer to the required decimal places, like four in this case, is essential for alignment with the problem's requirements. This ensures precision in your solutions, which is often necessary in academic work.