91Ó°ÊÓ

The sine function is a fundamental component of trigonometry. It is defined as the ratio of the length of the side opposite the angle to the hypotenuse in a right-angled triangle.
In the context of the unit circle, the sine function takes an angle as input and returns a value that represents the y-coordinate of the point on the circle corresponding to that angle.
This y-coordinate value ranges between -1 and 1. Here are some key properties:

• The sine of 0° (or 0 radians) is 0.
• At 90° (or Ï€/2 radians), the sine value peaks at 1.
• The sine of 180° (or Ï€ radians) returns to 0.
• At 270° (or 3Ï€/2 radians), the sine value reaches -1.

What this means is that the sine function goes through a full cycle every 360° (or 2π radians), repeating its values in predictable patterns. Understanding these properties helps in analyzing angles beyond the basic 90° increments, like with multiples and sums of π, such as in this exercise.

The unit circle is an essential concept in trigonometry, especially when working with trigonometric functions like the sine function.
A unit circle is a circle with a radius of 1, centered at the origin (0,0) in the xy-plane.
Each point on the unit circle can be defined using an angle, measured in radians, from the positive x-axis.

• The unit circle allows you to visualize the behavior of trigonometric functions, showing how the sine value (y-coordinate) and the cosine value (x-coordinate) change with angle.
• Angles such as \(\frac{\pi}{6}\) (30°), \(\frac{4\pi}{3}\) (240°), and \(\frac{5\pi}{2}\) (450°, or 90° after considering cycles) can be easily interpreted using the unit circle.

In this exercise, knowing these angles on the unit circle helps calculate function values without a calculator by identifying their corresponding sine values. For example, \(\sin\left(\frac{4\pi}{3}\right)\) equals -\(\sqrt{3}/2\) because at 240°, the point on the unit circle is in the third quadrant where the y-coordinate is negative.

Adding angles is a key operation in trigonometry, and it directly ties into understanding functions like sine and cosine.
In this exercise, we encounter angle addition when calculating \(\sin\left(\frac{4\pi}{3} + \frac{\pi}{6}\right)\).
The angle addition formula for sine is crucial:

• The formula states that \(\sin(a + b) = \sin(a)\cos(b) + \cos(a)\sin(b)\).
• In our case, adding \(\frac{4\pi}{3}\) and \(\frac{\pi}{6}\) simplifies the process, eventually showing that the angle wraps around the circle due to its larger-than-one full circle measurement.

This illustrates another key trigonometric principle: angles that exceed 2π or 360° repeat their sine values due to the periodic nature of these functions.
Understanding and applying angle addition formulas helps solve more complex trigonometric expressions effectively.