Problem 23 - Precalculus Mathematics for Calculus

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: Determine the Reference Angle

The angle \( \frac{2\pi}{3} \) is in the second quadrant. To find the reference angle \( \theta \), use \( \pi - \frac{2\pi}{3} \). This simplifies to \( \frac{\pi}{3} \).

Step 2: Evaluate Sine of Reference Angle

The sine of the reference angle \( \frac{\pi}{3} \) is known from the unit circle or trigonometric tables to be \( \frac{\sqrt{3}}{2} \).

Step 3: Determine the Sine Sign in the Second Quadrant

Since \( \sin \theta \) is positive in the second quadrant, the sine value of \( \frac{2\pi}{3} \) is the same as its reference angle.

Step 4: State the Exact Value

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

Key Concepts

Trigonometric Functions

Trigonometric functions are essential in mathematics to describe relationships within right-angled triangles. They relate angles with ratios of sides, primarily in the context of a right triangle, and are foundational for understanding circular motion. The primary trigonometric functions are the sine, cosine, and tangent, often abbreviated as sin, cos, and tan. Each trigonometric function has unique properties and graphs, which are periodic and repeat every full rotation (360 degrees or \(2\pi\) radians).

In this exercise, we focused on the sine function, which compares the length of the side opposite the angle to the hypotenuse in a right triangle. The sine function is periodic with a period of \(2\pi\), and it takes on values between -1 and 1.

Reference Angle

A reference angle is the smallest angle formed between the terminal side of the given angle and the x-axis. It helps determine the values of trigonometric functions for angles that are not in the first quadrant. This concept is vital as it transforms complex angles into simpler ones, allowing easier computation.

To find the reference angle for \(\frac{2\pi}{3}\), which lies in the second quadrant, subtract its angle from \(\pi\). This calculation gives us a reference angle of \(\frac{\pi}{3}\). This reference angle form is crucial because the trigonometric values are consistent across different quadrants, except for changes in sign.

Unit Circle

The unit circle is a circle with a radius of 1, centered at the origin of a coordinate plane. It is a fundamental tool for understanding how trigonometric functions behave. Each point on the circle corresponds to an angle formed with the positive x-axis, and using this, you derive the sine and cosine values as y and x coordinates, respectively.

Using the unit circle, the angle \(\frac{\pi}{3}\) (or 60 degrees) gives us a sine value of \(\frac{\sqrt{3}}{2}\). This is due to the coordinates of this angle on the unit circle being \((\frac{1}{2}, \frac{\sqrt{3}}{2})\), where the y-coordinate corresponds to the sine value. Knowing these coordinates is helpful for quickly solving angle-related trigonometry problems.

Second Quadrant

Angles located in the second quadrant range from \(\frac{\pi}{2}\) to \(\pi\) radians (90 to 180 degrees). In this quadrant, the sine function is positive, while the cosine function is negative. This is because of the unit circle's layout, where the sine value (y-coordinate) stays above zero in the second quadrant.

For the original exercise solving for \(\sin \frac{2\pi}{3}\), the second quadrant trait of a positive sine is pivotal. Although the reference angle \(\frac{\pi}{3}\) produces the same sine value \(\frac{\sqrt{3}}{2}\), its positivity remains because the sine function is positive in that quadrant. Understanding the behavior of trigonometric functions concerning quadrants simplifies finding exact values efficiently.