Problem 26 - Algebra and Trigonometry

Question

Find the exact value of the trigonometric function. $$ \cos \frac{7 \pi}{3} $$

Step-by-Step Solution

Answer

\( \cos \frac{7\pi}{3} = \frac{1}{2} \)

Step 1: Identify the Reference Angle

The given angle is \( \frac{7\pi}{3} \). To find a reference angle, note that one full rotation in radians is \( 2\pi \). Find a coterminal angle by subtracting \( 2\pi \):\[ \frac{7\pi}{3} - 2\pi = \frac{7\pi}{3} - \frac{6\pi}{3} = \frac{\pi}{3}. \] Thus, the reference angle is \( \frac{\pi}{3} \).

Step 2: Determine the Quadrant

After subtracting \( 2\pi \), the angle \( \frac{7\pi}{3} \) becomes \( \frac{\pi}{3} \), which lies in the first quadrant. In the first quadrant, all trigonometric functions are positive.

Step 3: Calculate the Cosine Value

Now, calculate the cosine of the reference angle \( \frac{\pi}{3} \). From the unit circle or trigonometric tables, we know that \( \cos \frac{\pi}{3} = \frac{1}{2} \). Since the angle is in the first quadrant, the cosine value remains positive.

Key Concepts

Reference Angle

A reference angle is a helpful tool in trigonometry that simplifies the process of evaluating trigonometric functions of larger angles. It is the smallest angle that a given angle shares with the x-axis. To find the reference angle of any angle in radians, you can subtract the closest lower complete rotations of \(2\pi\) (a full circle). For example, to find the reference angle of \( \frac{7\pi}{3} \):

• Recognize that \( 7\pi/3\) is more than \( 2\pi\), so it surpasses one full rotation.
• Calculate a coterminal angle by subtracting \( 2\pi\): \[ \frac{7\pi}{3} - 2\pi = \frac{7\pi}{3} - \frac{6\pi}{3} = \frac{\pi}{3}. \]
• This result, \( \frac{\pi}{3} \), is the reference angle.

Understanding reference angles allows us to focus on a familiar right triangle problem, rather than dealing with larger angles.

Unit Circle

The unit circle is an essential concept in trigonometry that helps you visualize and solve trigonometric problems easily. Essentially, the unit circle is a circle with a radius of 1 centered at the origin of a coordinate plane. It is the foundation for understanding trigonometric functions.In the unit circle:

• The angle is measured starting from the positive x-axis, moving counterclockwise.
• The coordinates of any point on the circle represent the cosine and sine values: (cos \(\theta\), sin \(\theta\)).

In our exercise, once you have the reference angle \(\frac{\pi}{3}\), you can find its position on the unit circle. In the unit circle:

• \(\frac{\pi}{3}\) is located in the first quadrant.
• As both cosine and sine are positive in this quadrant, identifying cosine is straightforward.

The unit circle allows us to quickly find trigonometric values for angles, especially when using well-known reference angles like \(\frac{\pi}{3}\).

Cosine Value

Finding the cosine value involves using the reference angle and the unit circle. For the trigonometric exercise given, understanding that the reference angle is \(\frac{\pi}{3}\) simplifies the calculation.Key details about the cosine function include:

• Cosine values can be quickly determined using special angles such as: \(0, \frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{3}, \frac{\pi}{2}\).
• For \(\frac{\pi}{3}\), the cosine value is known from trigonometric tables to be \(\frac{1}{2}\).
• Since \(\frac{\pi}{3}\) is in the first quadrant, the cosine value is positive.

Thanks to the property's simplicity and the unit circle, cosine values are accessible and easy to use in problems. The key step is understanding how to locate and interpret them correctly using the reference angle.