Problem 3 - Precalculus

Question

Find the exact value of the cosine and sine of the given angle. $$ \theta=\frac{\pi}{3} $$

Step-by-Step Solution

Answer

For \( \theta = \frac{\pi}{3} \), \( \cos\left(\frac{\pi}{3}\right) = \frac{1}{2} \) and \( \sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2} \).

Step 1: Identify the Angle

First, recognize that the given angle is \( \theta = \frac{\pi}{3} \). This angle is equivalent to 60 degrees. Knowing this is helpful, as 60 degrees is commonly found on the unit circle.

Step 2: Calculate Cosine

Cosine of an angle refers to the x-coordinate of the point on the unit circle at that angle. At \( \theta = \frac{\pi}{3} \), the cosine value is known to be \( \cos\left(\frac{\pi}{3}\right) = \frac{1}{2} \).

Step 3: Calculate Sine

Sine of an angle refers to the y-coordinate of the point on the unit circle at that angle. At \( \theta = \frac{\pi}{3} \), the sine value is known to be \( \sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2} \).

Key Concepts

Unit Circle

The unit circle is a circle with a radius of 1, centered at the origin of a coordinate plane. It's a vital tool in trigonometry for understanding how sine and cosine functions work. The circle’s key feature is that any angle \(\theta\) corresponds to a point on the circle, represented by coordinates \( (x, y) \). These coordinates are \( ( ext{cos} \theta, ext{sin} \theta) \).

• Radius: 1
• Center: Origin (0, 0)
• Coordinates: \( (\cos \theta, \sin \theta) \)

The unit circle allows for a geometric interpretation of trigonometric functions. Each angle, given in radians or degrees, corresponds to a point on this circle. For example, an angle of \(\frac{\pi}{3}\) radians, or 60 degrees, is located at a specific point on the circle.
By tracing the circle, you can understand how the sine and cosine values change with the angle. When you go around the circle, these values represent vertical and horizontal distances from the center, relating directly to the functions of sine and cosine.

Sine Function

The sine function is one of the primary trigonometric functions. It represents the vertical coordinate of a point on the unit circle at a given angle \(\theta\). Imagine you've chosen an angle \(\theta = \frac{\pi}{3}\), which is equivalent to 60 degrees, on the unit circle.

• Sine Formula: \( y = \sin \theta \)
• At \(\theta = \frac{\pi}{3}\): \(\sin \left( \frac{\pi}{3} \right) = \frac{\sqrt{3}}{2} \)

This function helps determine the y-coordinate on the unit circle, which corresponds to how "high" the point is from the horizontal axis. For the angle of 60 degrees, the altitude or height from the center to the edge of the circle is \(\frac{\sqrt{3}}{2}\), indicating the sine value.
Sine is periodic, which means it repeats its values in a regular cycle as you move around the circle. This makes it fundamental to the study of waves and oscillatory motion.

Cosine Function

The cosine function, closely related to sine, measures the horizontal part of the unit circle's angle. Specifically, it describes the x-coordinate at the given angle \(\theta\). For our angle \(\theta = \frac{\pi}{3}\), which equals 60 degrees, the cosine function gives us vital information.

• Cosine Formula: \( x = \cos \theta \)
• At \(\theta = \frac{\pi}{3}\): \(\cos \left( \frac{\pi}{3} \right) = \frac{1}{2} \)

Here, \(\frac{1}{2}\) is the x-coordinate for the point corresponding to 60 degrees on the unit circle, reflecting how far "across" from the y-axis the point is.
Cosine also has a periodic nature, repeating every \(2\pi\) radians, which means it's essential for understanding cycles and harmonic expressions in mathematics and physics. Similar to sine, it aids in modeling alternating phenomena like sound waves or electromagnetic cycles.