Problem 103 - Precalculus

Question

For the following exercises, state the reference angle for the given angle. $$ \frac{2 \pi}{3} $$

Step-by-Step Solution

Answer

The reference angle of \( \frac{2\pi}{3} \) is \( \frac{\pi}{3} \).

Step 1: Understand the concept of a reference angle

A reference angle is the smallest angle between the terminal side of the given angle and the x-axis. It is always a positive angle and can range between 0 and \( \frac{\pi}{2} \) radians.

Step 2: Identify the given angle's quadrant

The given angle is \( \frac{2\pi}{3} \). Since \( \frac{2\pi}{3} = \frac{\pi}{3} \times 2 \), this angle lies in the second quadrant because \( \frac{\pi}{2} < \frac{2\pi}{3} < \pi \).

Step 3: Calculate the reference angle

In the second quadrant, the reference angle is given by \( \pi - \theta \), where \( \theta \) is the given angle. Therefore, the reference angle is \( \pi - \frac{2\pi}{3} \).

Step 4: Simplify the expression

Simplify the expression \( \pi - \frac{2\pi}{3} \):\[ \pi - \frac{2\pi}{3} = \frac{3\pi}{3} - \frac{2\pi}{3} = \frac{\pi}{3} \].

Key Concepts

Angle Measurement

When measuring angles, it helps to understand the different systems and units involved. In mathematics, angles are often measured in degrees or radians. Degrees (\(°\)) are a familiar unit, where a complete circle is 360°. Radians (\(\pi\)) are another unit used widely in calculus and trigonometry, based on the radius of a circle. A full circle is \(2\pi\) radians. When working with trigonometric functions, radians provide a more natural way to describe angles in terms of \(\pi\). This makes them ideal for mathematical computations and theoretical work.
Understanding how to convert between these units is essential as well. To convert an angle in degrees to radians, use the formula:

• \[\text{radians} = \text{degrees} \times \frac{\pi}{180} \]

Conversely, to convert radians to degrees, the formula is:

• \[\text{degrees} = \text{radians} \times \frac{180}{\pi} \]

In this problem, we're dealing with radians directly. Recognizing that \(\frac{2\pi}{3}\) is in the radian system helps us correctly identify which quadrant an angle resides in.

Trigonometric Functions

Trigonometric functions like sine, cosine, and tangent provide values that represent the ratios of sides in a right triangle or the unit circle. These functions are crucial for understanding angles and their relationships. Each function corresponds to a specific pair of sides:

• **Sine (\(\sin\))**: Opposite side over Hypotenuse.
• **Cosine (\(\cos\))**: Adjacent side over Hypotenuse.
• **Tangent (\(\tan\))**: Opposite side over Adjacent side.

Additionally, these functions repeat their values over specific intervals, which is why we often refer to the **"circular"** nature of trigonometric functions. For example, the sine and cosine functions have a period of \(2\pi\), meaning their values repeat every \(2\pi\) radians. The cotangent, secant, and cosecant are related functions that also play a role in computations. Knowing the reference angle for a given angle allows us to determine the trigonometric values more straightforwardly since the reference angle gives us the same values of sine, cosine, and tangent, ignoring the potential sign change.

Quadrants

The coordinate plane is divided into four distinct quadrants, each with specific characteristics that affect angle measurements and trigonometric values. These quadrants are organized as follows:

• **First Quadrant**: Angles from \(0\) to \(\frac{\pi}{2}\). All trigonometric functions are positive.
• **Second Quadrant**: Angles from \(\frac{\pi}{2}\) to \(\pi\). Sine is positive, while cosine and tangent are negative.
• **Third Quadrant**: Angles from \(\pi\) to \(\frac{3\pi}{2}\). Tangent is positive, but sine and cosine are negative.
• **Fourth Quadrant**: Angles from \(\frac{3\pi}{2}\) to \(2\pi\). Cosine is positive, while sine and tangent are negative.

In this exercise, the angle \(\frac{2\pi}{3}\) falls in the second quadrant. Knowing this helps determine the angle's reference angle and predict the signs of trigonometric functions. To find the reference angle, we subtract the given angle from \(\pi\), which simplifies our calculations and allows for precise trigonometric evaluations for that segment of the unit circle.