Problem 120 - Precalculus

Question

For the following exercises, find the reference angle, the quadrant of the terminal side, and the sine and cosine of each angle. If the angle is not one of the angles on the unit circle, use a calculator and round to three decimal places. $$ \frac{4 \pi}{3} $$

Step-by-Step Solution

Answer

Reference angle is \( \frac{\pi}{3} \), it is in the third quadrant. \( \sin\left(\frac{4\pi}{3}\right) = -\frac{\sqrt{3}}{2}\), \( \cos\left(\frac{4\pi}{3}\right) = -\frac{1}{2} \).

Step 1: Find the Reference Angle

Start by determining the reference angle for \( \frac{4\pi}{3} \). Since \( \frac{4\pi}{3} \) is more than \( \pi \) but less than \( 2\pi \), it's in the third quadrant. The reference angle, which is the acute angle formed with the x-axis, is found by subtracting \( \pi \) from \( \frac{4\pi}{3} \): \[ \text{Reference Angle} = \frac{4\pi}{3} - \pi = \frac{4\pi}{3} - \frac{3\pi}{3} = \frac{\pi}{3}. \]

Step 2: Determine the Quadrant

Identify the quadrant where the terminal side of \( \frac{4\pi}{3} \) lies. Because \( \frac{4\pi}{3} \) is more than \( \pi \) but less than \( \frac{3\pi}{2} \), it lies in the third quadrant.

Step 3: Calculate Sine and Cosine

Using the reference angle \( \frac{\pi}{3} \) and knowing that in the third quadrant both sine and cosine are negative, compute: \[ \sin\left(\frac{4\pi}{3}\right) = -\sin\left(\frac{\pi}{3}\right) = -\frac{\sqrt{3}}{2}, \] \[ \cos\left(\frac{4\pi}{3}\right) = -\cos\left(\frac{\pi}{3}\right) = -\frac{1}{2}. \]

Key Concepts

Reference Angle

In trigonometry, the reference angle is an important concept that helps us simplify the evaluation of trigonometric functions. The reference angle is the smallest angle between the terminal side of a given angle and the x-axis. It is always a positive acute angle, meaning it is always less than or equal to 90 degrees or less than or equal to \( \frac{\pi}{2} \) radians.
To find the reference angle for an angle like \( \frac{4\pi}{3} \), determine which quadrant the angle is in and calculate the reference angle appropriately:

• If the angle lies in the first quadrant, the reference angle is the angle itself.
• For the second quadrant, subtract the angle from \( \pi \).
• In the third quadrant, subtract \( \pi \) from the angle.
• In the fourth quadrant, subtract the angle from \( 2\pi \).

As we saw, the reference angle for \( \frac{4\pi}{3} \) is \( \frac{\pi}{3} \). This concept is very useful when determining the sine and cosine values since these values are the same as those for the reference angle, but possibly with different signs depending on the quadrant.

Unit Circle

The unit circle is a circle with a radius of 1, centered at the origin of the coordinate plane. It plays a crucial role in trigonometry, especially when dealing with angles and their trigonometric functions values.
Any point on the unit circle can be represented as \((\cos \theta, \sin \theta)\), where \( \theta \) is the angle formed with the positive x-axis. The unit circle helps us understand the behavior of sine and cosine functions because:

• Sine corresponds to the y-coordinate of the point.
• Cosine corresponds to the x-coordinate of the point.

For angles like \( \frac{4\pi}{3} \), we can use their position on the unit circle to determine the trigonometric functions. The coordinates related to the reference angle, \( \frac{\pi}{3} \), are important; knowing in which quadrant the angle \( \frac{4\pi}{3} \) is will decide the sign of these coordinates.

Sine and Cosine Functions

Sine and cosine are fundamental trigonometric functions defined as the y and x coordinates of a point on the unit circle where a terminal side of an angle intersects. The sine function is written as \( \sin \theta \) and cosine as \( \cos \theta \).
Here are key properties of these functions:

• Sine and cosine values are between -1 and 1.
• They are periodic functions with a period of \( 2\pi \).

For the angle \( \frac{4\pi}{3} \), the reference angle \( \frac{\pi}{3} \) has known sine and cosine values. The actual values for \( \sin \left(\frac{4\pi}{3}\right) \) and \( \cos \left(\frac{4\pi}{3}\right) \) are equivalent to \( -\sin \left(\frac{\pi}{3}\right) = -\frac{\sqrt{3}}{2} \) and \( -\cos \left(\frac{\pi}{3}\right) = -\frac{1}{2} \), respectively, due to being in the third quadrant.

Quadrants in Trigonometry

The coordinate plane is divided into four quadrants, each affecting the sign of trigonometric functions. Knowing which quadrant an angle lies in helps determine the signs of its sine, cosine, and tangent values:

• First Quadrant (0 to \( \frac{\pi}{2} \)): Both sine and cosine are positive.
• Second Quadrant (\( \frac{\pi}{2} \) to \( \pi \)): Sine is positive, cosine is negative.
• Third Quadrant (\( \pi \) to \( \frac{3\pi}{2} \)): Both sine and cosine are negative.
• Fourth Quadrant (\( \frac{3\pi}{2} \) to \( 2\pi \)): Sine is negative, cosine is positive.

When dealing with an angle like \( \frac{4\pi}{3} \), it lies in the third quadrant. This means both sine and cosine values will be negative, which explains why the sine is \( -\frac{\sqrt{3}}{2} \) and cosine is \( -\frac{1}{2} \). Understanding the quadrant system is crucial for correctly determining the signs of trigonometric functions.