Problem 3 - Algebra and Trigonometry

Question

\(\sin \frac{2 \pi}{3}=\)________; \(\cos \frac{4 \pi}{3}=\) ________.

Step-by-Step Solution

Answer

\(\sin \frac{2\pi}{3} = \frac{\sqrt{3}}{2}\); \(\cos \frac{4\pi}{3} = -\frac{1}{2}\).

Step 1: Understanding the Unit Circle

The unit circle is a circle with a radius of 1 centered at the origin of the coordinate system. Angles measured in radians can help in identifying the values of sine and cosine for various standard positions on this circle.

Step 2: Locate \(\frac{2\pi}{3}\) on the Unit Circle

\(\frac{2\pi}{3}\) radians is equivalent to \(120^\circ\). This angle is in the second quadrant, where the sine function is positive and the cosine function is negative.

Step 3: Determine \(\frac{4\pi}{3}\) on the Unit Circle

\(\frac{4\pi}{3}\) radians is equivalent to \(240^\circ\). This angle is in the third quadrant, where both the sine and cosine functions are negative.

Step 4: Calculate \(\frac{2\pi}{3}\)

In the second quadrant, \(\frac{2\pi}{3}\) has a reference angle of \(\pi/3\), for which the sine value is \(\sin \frac{\pi}{3} = \frac{\sqrt{3}}{2}\).

Step 5: Calculate \(\frac{4\pi}{3}\)

In the third quadrant, \(\frac{4\pi}{3}\) has a reference angle of \(\pi/3\), for which the cosine value is \(\cos \frac{\pi}{3} = \frac{1}{2}\), but negative.

Step 6: Apply the Sine and Cosine Rules for Quadrants

For \(\frac{2\pi}{3}\), \(\sin \frac{2\pi}{3} = \frac{\sqrt{3}}{2}\) because sine is positive in the second quadrant. For \(\frac{4\pi}{3}\), \(\cos \frac{4\pi}{3} = -\frac{1}{2}\) because cosine is negative in the third quadrant.

Key Concepts

Sine Function

The sine function, often written as \(\text{sin}(x)\), measures the y-coordinate of a point on the unit circle corresponding to an angle x. Imagine the unit circle with a radius of 1. As you move around the circle, the sine function varies between -1 and 1. When you locate an angle on the unit circle, drop a perpendicular line to the x-axis. The length of this line is the value of the sine function. For example, to find \(\text{sin}(\frac{2\pi}{3})\), you look at the y-coordinate of the point on the unit circle at this angle, which is \(\frac{\text{√3}}{2}\).
The sine function is:

• Positive in Quadrants I and II
• Negative in Quadrants III and IV

Cosine Function

The cosine function, often represented as \( \text{cos}(x) \), measures the x-coordinate of a point on the unit circle corresponding to an angle x. Similar to the sine function, the cosine value varies between -1 and 1. When you measure an angle on the unit circle, drop a perpendicular line to the y-axis. The length of this line is the value of the cosine function. For instance, to find \( \text{cos}(\frac{4\text{π}}{3}) \), you look at the x-coordinate of the point on the unit circle at this angle, which is \(-\frac{1}{2}\).
The cosine function is:

• Positive in Quadrants I and IV
• Negative in Quadrants II and III

Reference Angles

Reference angles are the acute angles (less than 90 degrees) that a given angle makes with the x-axis. These angles help simplify calculations for sine and cosine. To find a reference angle, imagine the smallest angle between the terminal side of the angle and the x-axis. For example, the reference angle for \( \frac{2\text{π}}{3} \) is \( \frac{π}{3} \), because this angle lies in the second quadrant. To use a reference angle effectively:

• Identify the angle's quadrant
• Subtract the known angle from \( π \), \( 2π \), or \( \frac{3π}{2} \) based on its position
• Apply sine and cosine values for the reference angle

Researching \( \frac{4\text{π}}{3} \), you subtract \( π \) from it to get the reference angle, which is again \( \frac{π}{3} \).

Quadrants

The unit circle is divided into four quadrants, each affecting the sign and value of the sine and cosine functions:

• Quadrant I: Angles from \( 0 \) to \( \frac{π}{2} \), both sine and cosine are positive.
• Quadrant II: Angles from \( \frac{π}{2} \) to \( π \), sine is positive and cosine is negative.
• Quadrant III: Angles from \( π \) to \( \frac{3π}{2} \), both sine and cosine are negative.
• Quadrant IV: Angles from \( \frac{3π}{2} \) to \( 2π \), sine is negative and cosine is positive.

Understanding in which quadrant an angle lies helps to determine the signs of the sine and cosine values, ensuring accurate calculations. For example, \( \frac{2\text{π}}{3} \) lies in Quadrant II, where the sine is positive and the cosine is negative. In contrast, \( \frac{4\text{π}}{3} \) is in Quadrant III, where both sine and cosine are negative.