Problem 7 - Algebra 2 and Trigonometry

Question

In \(3-12\) , find the exact function value of each of the following if the measure of the angle is given in radians. $$ \cos \frac{2 \pi}{3} $$

Step-by-Step Solution

Answer

The value of \( \cos \frac{2\pi}{3} \) is \(-\frac{1}{2}\).

Step 1: Identify the Reference Angle

The provided angle is \( \frac{2\pi}{3} \) radians. Since the angle is in the second quadrant (because \( \frac{2\pi}{3} \) is between \( \pi/2 \) and \( \pi \)), we need to find its reference angle. The reference angle \( \theta_r \) can be calculated as \( \pi - \frac{2\pi}{3} = \frac{\pi}{3} \).

Step 2: Determine the Sign of Cosine in the Second Quadrant

In the second quadrant, the cosine function is negative, because the x-values are negative while the radius of the unit circle is always positive. Hence, cosine is negative in this quadrant.

Step 3: Use the Known Value of Cosine of the Reference Angle

The reference angle \( \frac{\pi}{3} \) corresponds to one of the standard angles on the unit circle. The cosine of \( \frac{\pi}{3} \) is \( \frac{1}{2} \).

Step 4: Apply the Sign for Second Quadrant

Since we are in the second quadrant and cosine is negative here, the cosine for \( \frac{2\pi}{3} \) is negative. Thus, \( \cos \frac{2\pi}{3} = -\frac{1}{2} \).

Key Concepts

Radians

In trigonometry, the concept of radians is essential. Unlike degrees, which divide a circle into 360 parts, radians are based on the arc length of the circle. This measurement system captures the relationship between the radius of a circle and its circumference.
One full rotation around a circle equals \( 2\pi \) radians, or roughly \( 6.283 \) radians. Therefore, a half-circle (or \( \pi \) radians) equals \( 180 \) degrees, and \( 1 \) radian is approximately \( 57.3 \) degrees.
When angles are given in radians, calculations involving trigonometric functions can often be more straightforward. In some exercises, angles are provided in fractions of \( \pi \), making it easier to see their position on the unit circle.

Reference Angle

To find the function value of an angle given in radians, identifying the reference angle can greatly simplify the process. A reference angle is the smallest angle that a given angle forms with the x-axis. It is always positive and lies between \( 0 \) and \( \frac{\pi}{2} \) radians.
Here's how to find the reference angle based on the quadrant:

• First Quadrant: The reference angle is the angle itself.
• Second Quadrant: Subtract the angle from \( \pi \).
• Third Quadrant: Subtract \( \pi \) from the angle.
• Fourth Quadrant: Subtract the angle from \( 2\pi \).

For an angle of \( \frac{2\pi}{3} \), which is in the second quadrant, the reference angle is \( \pi - \frac{2\pi}{3} = \frac{\pi}{3} \). This simplification allows us to use known trigonometric values more easily.

Unit Circle

The unit circle is a vital tool for understanding trigonometric functions and their properties. It is a circle with a radius of 1, centered at the origin of the coordinate plane. This setup makes it easier to visualize the angle measures and the lengths of sine, cosine, and other trigonometric functions.
Angles on the unit circle are typically measured in radians. The x-coordinate represents the cosine of the angle, while the y-coordinate represents the sine of the angle.
If you imagine the angle \( \frac{2\pi}{3} \) on the unit circle, its reference angle \( \frac{\pi}{3} \) has a known cosine: \( \frac{1}{2} \). Because it's in the second quadrant, cosine becomes negative, resulting in \( \cos \frac{2\pi}{3} = -\frac{1}{2} \). The unit circle makes such calculations visible and comprehensible.

Quadrants

The Cartesian coordinate system, divided into four quadrants, plays a crucial role in trigonometry. Each quadrant has distinct characteristics that affect the sign of trigonometric functions.
- Quadrant I: Both x and y coordinates are positive, so sine and cosine are positive.- Quadrant II: x is negative, y is positive, so sine is positive, cosine is negative.- Quadrant III: Both x and y coordinates are negative, so sine and cosine are negative.- Quadrant IV: x is positive, y is negative, so sine is negative, cosine is positive.The angle \( \frac{2\pi}{3} \) is located in the second quadrant where cosine values are negative. Therefore, knowing the quadrant helps determine the sign of the trigonometric function, as seen in exercises where \( \cos \frac{2\pi}{3} = -\frac{1}{2} \). Identifying the quadrant is an essential step in simplifying problems involving angles and trigonometric functions.