Use the unit circle shown to find the value of the trigonometric function.
sin (2𝜋/3)

Verified step by step guidance

1. Step 1: Recognize that the angle $\frac{2\pi }{3}$ is in radians and corresponds to 120 degrees.
2. Step 2: Identify the position of $\frac{2\pi }{3}$ on the unit circle. It is in the second quadrant.
3. Step 3: Recall that in the second quadrant, the sine function is positive.
4. Step 4: Use the reference angle for $\frac{2\pi }{3}$, which is $\pi -\frac{2\pi }{3}=\frac{\pi }{3}$.
5. Step 5: The sine of $\frac{2\pi }{3}$ is the same as the sine of its reference angle $\frac{\pi }{3}$, which is $\sin (\frac{\pi }{3})$.

Key Concepts

Unit Circle

The unit circle is a circle with a radius of one centered at the origin of a coordinate plane. It is a fundamental tool in trigonometry, as it allows for the definition of trigonometric functions based on angles measured from the positive x-axis. Each point on the unit circle corresponds to a specific angle and provides the sine and cosine values for that angle, which are essential for evaluating trigonometric functions.

Sine Function

The sine function, denoted as sin(θ), is a trigonometric function that represents the y-coordinate of a point on the unit circle corresponding to an angle θ. For angles in the second quadrant, such as 2π/3, the sine value is positive. Understanding the sine function's behavior in different quadrants is crucial for accurately determining its value for various angles.

Angle Measurement in Radians

In trigonometry, angles can be measured in degrees or radians, with radians being the standard unit in mathematical contexts. The angle 2π/3 radians corresponds to 120 degrees, placing it in the second quadrant of the unit circle. Recognizing how to convert between degrees and radians and understanding the implications of angle placement on the unit circle is vital for evaluating trigonometric functions.