Problem 38 - Precalculus

Question

Use a calculator to evaluate the trigonometric functions for the indicated angle values. Round your answers to four decimal places. $$\cos \left(\frac{13 \pi}{7}\right)$$

Step-by-Step Solution

Answer

\( \cos(\frac{13\pi}{7}) \approx 0.6235 \)

Step 1: Understand the Angle in Radians

The given angle is \( \frac{13\pi}{7} \) radians. This is slightly less than \( 2\pi \) since \( 2\pi \approx 6.283 \) and \( 13\pi/7 \approx 5.833 \). This means the angle is in the fourth quadrant of the unit circle.

Step 2: Use a Calculator

Using a scientific calculator, enter the trigonometric function for cosine and input \( \frac{13\pi}{7} \). Ensure the calculator is in radian mode, as the angle is provided in radians.

Step 3: Calculate Cosine Value

After entering \( \cos(\frac{13\pi}{7}) \) into the calculator, calculate the result. Make note of the precision and round the result to four decimal places.

Step 4: Check the Quadrant and Sign

Since the angle is in the fourth quadrant, the cosine of this angle should be positive. Verify that your calculator output is positive, which aligns with the expected sign for cosine in the fourth quadrant.

Key Concepts

Radians

Radians are a way to measure angles. Instead of using degrees, which split a circle into 360 parts, radians use the radius of the circle. One complete circle is equal to \( 2\pi \) radians. This means that \( \pi \) radians is half a circle, equivalent to 180 degrees.

Using radians can make calculations involving circles, arcs, and trigonometric functions simpler in mathematics. When working with trigonometric functions, it's important to ensure your calculator is set to "radian mode" to avoid incorrect results. In the given problem, the angle \( \frac{13\pi}{7} \) is provided in radians, so calculations should be done accordingly.

Unit Circle

The unit circle is a fundamental concept in trigonometry. It is a circle with a radius of 1, centered at the origin of a coordinate plane. By using the unit circle, we can easily understand the values of trigonometric functions like sine and cosine.

On the unit circle, each point can be represented by \((\cos \theta, \sin \theta)\), where \(\theta\) is the angle in radians measured from the positive x-axis. This setup makes the calculations stable and consistent. For an angle of \( \frac{13\pi}{7} \), the corresponding point on the unit circle helps in directly evaluating the cosine function.

Quadrants

The coordinate plane is divided into four sections known as quadrants. Starting from the positive x-axis and moving counterclockwise:

• Quadrant I: where both x and y-values are positive
• Quadrant II: where x-values are negative and y-values are positive
• Quadrant III: where both x and y-values are negative
• Quadrant IV: where x-values are positive and y-values are negative

Knowing in which quadrant an angle lies helps determine the sign of trigonometric functions. For instance, in Quadrant IV, the cosine is positive whereas the sine is negative. In our problem, since \( \frac{13\pi}{7} \) is less than \(2\pi\) but more than \(\pi\), it lies in Quadrant IV. This explains why the cosine value should be positive.

Cosine Function

The cosine function is a fundamental trigonometric function that represents the x-coordinate of a point on the unit circle at a given angle \(\theta\). Cosine values range from -1 to 1.

Cosine is periodic, with a period of \( 2\pi \), meaning \( \cos(\theta) = \cos(\theta + 2\pi n) \) for any integer \( n \). It's useful to identify angles within the \( 0 \) to \( 2\pi \) interval to easily compute cosine values. For the angle \( \frac{13\pi}{7} \), its cosine can be calculated using a calculator set to radian mode, and it's expected to yield a positive result since the angle is in the fourth quadrant. The resultant value needs to be rounded to four decimal places as per the exercise requirement.