Problem 17 - Precalculus

Question

Use the unit circle and the fact that sine is an odd function and cosine is an even function to find the exact values of the indicated functions. $$\sin \left(-\frac{\pi}{3}\right)$$

Step-by-Step Solution

Answer

\(\sin\left(-\frac{\pi}{3}\right) = -\frac{\sqrt{3}}{2}\).

Step 1: Understanding the Unit Circle

The unit circle is a circle with a radius of 1, centered at the origin of a coordinate plane. Important angles on the unit circle, like \(-\frac{\pi}{3}\), have specific coordinates that help determine sine and cosine values. Specifically, the point corresponding to \(-\frac{\pi}{3}\) must be reviewed based on its wave form nature and symmetry about the axes.

Step 2: Evaluating the Angle

To find \(\sin\left(-\frac{\pi}{3}\right)\), it's critical to note that \(-\frac{\pi}{3}\) corresponds to a rotation of \(60^\circ\) in the clockwise direction from the positive x-axis.

Step 3: Using the Property of Sine as an Odd Function

The sine function is odd, meaning \(\sin(-x) = -\sin(x)\). Thus, \(\sin\left(-\frac{\pi}{3}\right) = -\sin\left(\frac{\pi}{3}\right)\).

Step 4: Finding the Exact Value of Sine at \(\frac{\pi}{3}\)

The angle \(\frac{\pi}{3}\) on the unit circle has sine as the y-coordinate of the point \(\left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right)\). Therefore, \(\sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2}\).

Step 5: Applying the Odd Function Rule to Find the Final Result

Applying the rule from earlier, \(\sin\left(-\frac{\pi}{3}\right) = -\frac{\sqrt{3}}{2}\).

Key Concepts

Sine Function

The sine function is one of the key trigonometric functions and is integral to understanding trigonometry. When we talk about the sine function, we're referring to the ratio of the opposite side to the hypotenuse in a right triangle.
In the context of the unit circle, which is a circle with a radius of 1 centered on the origin, the sine of an angle is defined as the y-coordinate of the point where the terminal side of the angle intersects the circle.

• Sine values vary between -1 and 1, due to the unit circle's radius.
• In quadrants, sine is positive in the first and second, and negative in the third and fourth.

Understanding the sine function in terms of the unit circle helps visualize and compute sine values for various angles.

Odd Function

An odd function has a specific symmetry around the origin, such that the function satisfies the property: \(f(-x) = -f(x)\). The sine function is one of the most famous examples of an odd function.

• This property means that the graph of sine reflects perfectly across the origin.
• Understanding sine as an odd function allows us to compute negative angle values easily.

For example, if you want to find \( \sin(-\frac{\pi}{3}) \), you simply take the negative of \( \sin(\frac{\pi}{3}) \). Hence, \( \sin(-\frac{\pi}{3}) = -\frac{\sqrt{3}}{2} \).
This reflection property simplifies solving problems involving negative angles, which are common in trigonometric exercises.

Exact Trigonometric Values

Exact trigonometric values refer to specific, well-defined values for sine, cosine, and tangent at commonly used angles such as \(0, \frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{3}, \) and \( \frac{\pi}{2} \).
These values are derived from the properties of the 30°-60°-90° triangle and the 45°-45°-90° triangle.
Some key exact values via the unit circle are:

• \( \sin(\frac{\pi}{6}) = \frac{1}{2} \)
• \( \sin(\frac{\pi}{4}) = \frac{\sqrt{2}}{2} \)
• \( \sin(\frac{\pi}{3}) = \frac{\sqrt{3}}{2} \)

These exact values are important as they provide a convenient reference, simplifying calculations when these specific angles appear.
Using the unit circle, these values are helpful in practical applications, from simple geometry to engineering, where measuring accurate angles is crucial.