Problem 27 - Algebra and Trigonometry

Question

Find the exact value of the trigonometric function. $$ \cos \left(-\frac{7 \pi}{3}\right) $$

Step-by-Step Solution

Answer

\(\cos\left(-\frac{7\pi}{3}\right) = \frac{1}{2}\).

Step 1: Identify the angle

The given angle is \(-\frac{7\pi}{3}\). This angle is in radians.

Step 2: Convert the negative angle to a positive angle

Negative angles rotate clockwise. We convert \(-\frac{7\pi}{3}\) into a positive angle by adding \(2\pi\) (a full rotation): \[-\frac{7\pi}{3} + 2\pi = -\frac{7\pi}{3} + \frac{6\pi}{3} = -\frac{\pi}{3}.\]

Step 3: Find the equivalent angle within standard range

Since \(-\frac{\pi}{3}\) is still negative, add another \(2\pi = \frac{6\pi}{3}\) to reach an equivalent positive angle:\[-\frac{\pi}{3} + \frac{6\pi}{3} = \frac{5\pi}{3}.\] Now, \(\frac{5\pi}{3}\) is within \([0, 2\pi)\).

Step 4: Evaluate cosine of the angle within the unit circle

The angle \(\frac{5\pi}{3}\) is equivalent to \(2\pi - \frac{\pi}{3}\), which places it in the fourth quadrant. In the fourth quadrant, cosine is positive and \(\cos\left(\frac{5\pi}{3}\right) = \cos\left(-\frac{\pi}{3}\right) = \cos\left(\frac{\pi}{3}\right).\)

Step 5: Determine the cosine's exact value using known reference angles

The reference angle is \(\frac{\pi}{3}\), and from trigonometric values of special angles, \(\cos\left(\frac{\pi}{3}\right) = \frac{1}{2}\).

Key Concepts

Cosine and Its Importance

Understanding the cosine function is crucial when studying trigonometry. Cosine is one of the primary trigonometric functions and it measures the ratio of the adjacent side to the hypotenuse in a right-angled triangle. This function is essential because it helps us find unknown angles and lengths in various geometric problems.

Like all trigonometric functions, cosine can be defined for any angle, not just for acute angles in a triangle. It's often represented in the unit circle, which is vital to understanding how it behaves over different quadrants.

• For angles in the first quadrant, cosine values range from 1 to 0.
• In the second quadrant, cosine values are negative, reducing from 0 to -1.
• The third quadrant also sees negative cosine values, increasing from -1 back to 0.
• Finally, the fourth quadrant is where cosine values are positive again, increasing from 0 to 1.

This alternating pattern highlights the periodic nature of cosine, which repeats every full circle or \(2\pi\) radians.

Exploring the Unit Circle

The unit circle is an invaluable tool for understanding trigonometric functions, including cosine. It's a circle centered at the origin of a coordinate plane with a radius of 1. This simple diagram allows us to define trigonometric functions for all angles, not just those in right triangles.

In the unit circle, the cosine of an angle corresponds to the x-coordinate of the point where the terminal side of the angle intersects the circle. This method provides a visual means to grasp how cosine changes as angles vary.

• Every angle on the unit circle can be represented in radians, often simplifying calculations.
• Positive angles are measured counterclockwise from the positive x-axis, while negative angles are measured clockwise.
• The unit circle helps identify which quadrant an angle lies, crucial for determining the sign of cosine.

Overall, the unit circle is a powerful conceptual aid for visual learners aiming to master trigonometry concepts.

The Role of Reference Angles

Reference angles can simplify finding trigonometric functions of any given angles, especially those outside the typical \(0\) to \(2\pi\) radian span. A reference angle is always a positive acute angle, providing the trigonometric functions of angles in any quadrant.

To find a reference angle, determine the shortest angle back to the x-axis. By using reference angles, the known values for the standard angles can apply across quadrants.

• Each reference angle represents an angle's magnitude without regard to direction.
• In the exercise example, \(\frac{5\pi}{3}\) is simplified to its reference angle \(\frac{\pi}{3}\) to find \(\cos\left(\frac{\pi}{3}\right) = \frac{1}{2}\).
• Reference angles ensure answers are precise and quickly derived from memorized special angle values without recalculating complex angles.

Understanding reference angles enriches problem-solving skills and sharpens trigonometric intuition, making complex calculations more approachable.