Problem 23 - Algebra and Trigonometry

Question

Find the exact value of the trigonometric function. $$ \sin \frac{2 \pi}{3} $$

Step-by-Step Solution

Answer

\( \sin \frac{2\pi}{3} = \frac{\sqrt{3}}{2} \)

Step 1: Recognize the Angle in Terms of Common Angles

The angle \( \frac{2\pi}{3} \) is equivalent to 120 degrees. This means we're dealing with a familiar angle that falls into the second quadrant of the unit circle.

Step 2: Evaluate the Sine Function in the Second Quadrant

In the second quadrant, the sine of an angle is positive. The reference angle for \( \frac{2\pi}{3} \) is \( \pi - \frac{2\pi}{3} = \frac{\pi}{3} \), which corresponds to 60 degrees.

Step 3: Use Known Sine Values

The sine of \( 60^{\circ} \) or \( \frac{\pi}{3} \) is known to be \( \frac{\sqrt{3}}{2} \). Since the angle \( \frac{2\pi}{3} \) is in the second quadrant where sine is positive, the value remains \( \frac{\sqrt{3}}{2} \).

Step 4: Conclusion: Find the Exact Value

Thus, the exact value of \( \sin \frac{2\pi}{3} \) is \( \frac{\sqrt{3}}{2} \).

Key Concepts

Unit Circle

The unit circle is a fundamental concept in trigonometry, essential for understanding trigonometric functions like sine, cosine, and tangent. Imagine a circle with a radius of one unit centered at the origin of a coordinate plane. This is the unit circle.
On this circle, every point represents the endpoint of a radius extending from the origin at an angle \( \theta \). The coordinates of this point are \((\cos \theta, \sin \theta)\), making the unit circle a powerful tool for defining trigonometric functions.
The circle is divided into four quadrants, each corresponding to a range of angles. Each quadrant is important because the signs of the sine and cosine functions change depending on which quadrant you are in. This is why understanding the unit circle is crucial for solving trigonometric problems.

Sine Function

The sine function is one of the main trigonometric functions. Its value corresponds to the y-coordinate of the point on the unit circle at a given angle \( \theta \). For each angle \( \theta \), the sine function helps determine how far up or down a point is from the horizontal axis when projected onto the unit circle.
One of the key features of the sine function is its periodic nature, repeating every \( 2\pi \) radians, or 360 degrees. This means by adding or subtracting full rotations, the sine of an angle remains unchanged. The range of the sine function's output is from -1 to 1, which can be seen from the maximum and minimum y-values on the unit circle.
Knowing specific angles and their sine values is beneficial for quickly solving problems. For instance, the sine of \( \frac{\pi}{3} \) or 60 degrees is \( \frac{\sqrt{3}}{2} \), a common angle used in problems.

Quadrants of the Unit Circle

The unit circle is split into four quadrants, making it easier to determine the signs of trigonometric functions like sine and cosine. Each quadrant represents a distinct range of angles:

• First Quadrant (0 to \( \frac{\pi}{2} \) radians): Both sine and cosine are positive.
• Second Quadrant (\( \frac{\pi}{2} \) to \( \pi \) radians): Sine is positive, but cosine is negative.
• Third Quadrant (\( \pi \) to \( \frac{3\pi}{2} \) radians): Both sine and cosine become negative.
• Fourth Quadrant (\( \frac{3\pi}{2} \) to \( 2\pi \) radians): Sine is negative, while cosine is positive.

Understanding these quadrants is crucial as they help predict the sign of the sine and cosine functions for any angle. For example, the angle \( \frac{2\pi}{3} \) lies in the second quadrant. This tells us that the sine will be positive while cosine will be negative, influencing how we solve trigonometric equations or find exact values like in our original problem.