Problem 26 - Precalculus Mathematics for Calculus

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 and Simplify the Angle

The given angle is \( \frac{7 \pi}{3} \). This is more than \( 2\pi \), meaning it's in another rotation of the unit circle. To find an equivalent angle between \( 0 \) and \( 2\pi \), subtract \( 2\pi \) from \( \frac{7\pi}{3} \).

Step 2: Calculate Equivalent Angle

Subtract \( 2\pi \) from \( \frac{7\pi}{3} \) :\[ \frac{7\pi}{3} - 2\pi = \frac{7\pi}{3} - \frac{6\pi}{3} = \frac{\pi}{3}. \]So, \( \cos \frac{7\pi}{3} = \cos \frac{\pi}{3}. \)

Step 3: Find the Exact Value

The angle \( \frac{\pi}{3} \) is one of the standard angles in trigonometry. The cosine of \( \frac{\pi}{3} \) is \( \frac{1}{2} \).Thus, \( \cos \frac{7\pi}{3} = \frac{1}{2}. \)

Key Concepts

Understanding Cosine

Cosine is one of the primary trigonometric functions and it helps us understand relationships involving angles and lengths. When you think of cosine, imagine the adjacent side over the hypotenuse in a right-angled triangle: \[ \cos(\theta) = \frac{\text{adjacent side}}{\text{hypotenuse}}. \]
It's a fundamental tool in mathematics because it lets us translate between angle measures and ratios of length. In the unit circle, the cosine value of an angle gives us the x-coordinate of the corresponding point on the circle, which helps in further evaluating trigonometric functions.
Understanding cosine is crucial if you're dealing with circular motion, wave functions, or any field requiring trigonometry.

The Art of Angle Conversion

Angles in trigonometry can be measured in degrees or radians. Radians are more often used in higher mathematics because they relate directly to the unit circle. Conversion between these forms is essential. For any angle \( \theta \) measured in radians:

• The full circle is \( 2\pi \) radians.
• A straight angle is \( \pi \) radians.

Sometimes angles exceed \( 2\pi \) radians. In such cases, converting or reducing these angles into standard position (within \( 0 \) to \( 2\pi \) radians) is useful. For instance, \( \frac{7\pi}{3} \) radians can be simplified by subtracting \( 2\pi \) to bring it back within the first rotation around the circle. This approach makes handling angles much simpler and aligns them within the standard \( 0 \) to \( 2\pi \) range.
This keeps trigonometric calculations straightforward and ensures consistent results.

Exploring the Unit Circle

The unit circle is a circle drawn on a coordinate plane with a radius of one centered at the origin (0,0). It's a powerful concept in trigonometry because it provides a geometric framework for understanding angles and their trigonometric ratios.
On the unit circle, every point \((x, y)\) is defined by an angle \(\theta\) from the positive x-axis. The x-coordinate of each point is the cosine of \(\theta\), and the y-coordinate is the sine of \(\theta\). This makes the unit circle an incredible tool for visualizing trigonometric functions:

• Each angle corresponds to specific coordinates.
• The values of sine and cosine remain between \(-1 \) and \(1\).

Utilizing the unit circle, you can easily determine values of trigonometric functions at standard angles like \(\frac{\pi}{3}\) and \(\frac{\pi}{6}\).
This framework also helps in understanding periodic behaviors of functions, which are crucial in both mathematical theories and real-world applications.