Problem 14 - Precalculus : Building Concepts and Connections

Question

Skills This set of exercises will reinforce the skills illustrated in this section. In Exercises \(9-22,\) find the reference angle for each of the angles given. $$\frac{4 \pi}{3}$$

Step-by-Step Solution

Answer

The reference angle for \( \frac{4 \pi}{3}\) is \( \frac{\pi}{3}\)

Step 1: Identify the quadrant of the angle

The angle \( \frac{4 \pi}{3}\) lies in the third quadrant of the unit circle as \( \frac{3 \pi}{3} < \frac{4 \pi}{3} < \frac{5 \pi}{3}\)

Step 2: Find the reference angle

Because the given angle is in the third quadrant, the reference angle \( \theta_{r} \) is found by subtracting the given angle from \(\pi\). \n\n\(\theta_{r} = \pi - |\frac{4 \pi}{3}|\)\n\nThis simplifies to: \( \theta_{r} = \frac{\pi}{3}\)

Key Concepts

Unit Circle

The unit circle is a fundamental concept in trigonometry and mathematics. It is a circle with a radius of 1, centered at the origin of a coordinate plane. The beauty of the unit circle is its simplicity, which allows us to easily visualize and calculate trigonometric functions.Within this circle, angles are measured starting from the positive x-axis.

• Angles can be measured in radians or degrees, but radians are more commonly used in higher mathematics.
• Completing one full circle around the circle corresponds to an angle of either \(2\pi\) radians or 360 degrees.

The unit circle helps us map the angle values to coordinates on the circle, where the x-coordinate represents the cosine of the angle and the y-coordinate represents the sine of the angle.Understanding the unit circle helps us define trigonometric functions such as sin, cos, and tan, as well as understanding the concepts of reference angles and periodicity in trigonometry.

Radians

Radians are another way to measure angles, often preferable to degrees in mathematics because they offer a more natural unit for measuring angles based on the properties of circles. One radian is the angle formed when the arc length of a circle is equal to its radius.

• An important relation to remember is that \(2\pi\) radians is equal to 360 degrees, which implies \(\pi\) radians is equivalent to 180 degrees.
• This means each quadrant of the unit circle, which is 90 degrees, corresponds to \(\pi/2\) radians.

Why Radians Are UsefulRadians simplify various mathematical formulas and calculations, particularly in calculus, where they allow for the derivative of the sine function to be expressed as cosine, and vice versa, without additional constants. By using radians, the angle in question can often be more directly and simply incorporated into equations and functions, especially in periodic and oscillatory contexts.

Angles in Quadrants

The understanding of angles in different quadrants is crucial for identifying reference angles and solving trigonometric equations. The four quadrants of the unit circle separate the plane into four regions, each containing different angle measures.

• First Quadrant: Angles between \(0\) and \(\pi/2\) radians. Here, both sine and cosine values are positive.
• Second Quadrant: Angles between \(\pi/2\) and \(\pi\) radians. Sine values are positive, while cosine values are negative.
• Third Quadrant: Angles between \(\pi\) and \(3\pi/2\) radians. Both sine and cosine values are negative.
• Fourth Quadrant: Angles between \(3\pi/2\) and \(2\pi\) radians. Sine values are negative, while cosine values are positive.

For the original exercise, the angle \(\frac{4\pi}{3}\) is located in the third quadrant, meaning both its sine and cosine values are negative, as explained in the solution steps. Understanding these properties helps in finding the reference angle which is crucial for trigonometric identities and equations.