Problem 30 - Precalculus

Question

Use the unit circle and the fact that sine is an odd function and cosine is an even function to find the exact values of the indicated functions. $$\cos \left(-\frac{7 \pi}{6}\right)$$

Step-by-Step Solution

Answer

\(-\frac{\sqrt{3}}{2}\)

Step 1: Understanding Even Function Property

The cosine function is an even function. This means that for any angle \(x\), \( \cos(-x) = \cos(x) \). Here, the angle is \(-\frac{7\pi}{6}\). According to the even function property, \( \cos\left(-\frac{7\pi}{6}\right) = \cos\left(\frac{7\pi}{6}\right) \).

Step 2: Locate the Angle on the Unit Circle

\( \frac{7\pi}{6} \) is an angle in the third quadrant of the unit circle. It corresponds to \( 210^\circ \), as \( \pi \) radians equals \( 180^\circ \), and \( \frac{7\pi}{6} \times 180^\circ/\pi = 210^\circ \).

Step 3: Determine Reference Angle

The reference angle is the acute angle formed by the terminal side of \(\frac{7\pi}{6}\) and the x-axis. Since \(210^\circ - 180^\circ = 30^\circ\), the reference angle is \(30^\circ\) or \(\frac{\pi}{6}\).

Step 4: Use Cosine Value of Reference Angle

In the unit circle, \(\cos\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2}\). Since \(\frac{7\pi}{6}\) is in the third quadrant, where cosine values are negative, \(\cos\left(\frac{7\pi}{6}\right) = -\frac{\sqrt{3}}{2}\).

Step 5: Apply Even Function Characteristic

Finally, using that \(\cos\left(-\frac{7\pi}{6}\right) = \cos\left(\frac{7\pi}{6}\right)\), we conclude that \(\cos\left(-\frac{7\pi}{6}\right) = -\frac{\sqrt{3}}{2}\).

Key Concepts

Even Function

When talking about even functions in mathematics, we are referring to a very particular type of function with a specific symmetry. An even function is characterized by the fact that the function's value at a positive input is identical to its value at the corresponding negative input. This can be formally expressed as:

• If \( f(x) = f(-x) \), then \( f \) is an even function.

In the context of trigonometric functions, the cosine function is a classic example of an even function. This property simplifies calculations significantly since, for any angle \( x \), \( \cos(-x) = \cos(x) \).
This means you can reflect the angle over the y-axis without changing the cosine's value. It helps in solving problems by reducing unnecessary recalculations of function values for reflected angles.

Cosine

Cosine is a fundamental trigonometric function that you encounter often, especially when working with angles and triangles. Cosine corresponds to the x-coordinate of a point on the unit circle that is a certain angle away from the positive x-axis.

• For any angle \( x \), the point on the unit circle has coordinates \( (\cos(x), \sin(x)) \).

Using the unit circle helps visualize how the cosine function behaves for angles in different quadrants. Cosine values are positive in the first and fourth quadrants and negative in the second and third quadrants.
This understanding is crucial when applying the properties of even functions to find values like \( \cos\left(-\frac{7\pi}{6}\right) \). By seeing where the angle falls on the unit circle, you recognize the sign of the cosine based on the angle's quadrant position.

Reference Angle

A reference angle is a helpful concept used when working with angles not located in the first quadrant of the unit circle. The reference angle is the smallest possible angle between the terminal side of a given angle and the x-axis.

• The reference angle is always positive and measures between \(0\) and \(\frac{\pi}{2}\) radians, or \(0\) and \(90^{\circ}\).

For instance, when dealing with the angle \( \frac{7\pi}{6} \), which is found in the third quadrant, you calculate its reference angle by subtracting \(180^{\circ} \) from \(210^{\circ} \), resulting in \(30^{\circ} \) or \(\frac{\pi}{6} \).
Knowing the reference angle allows you to determine the cosine value by referring back to the known values of standard angles.

Third Quadrant

Understanding the third quadrant of the unit circle is vital when solving trigonometry problems involving angles like \( \frac{7\pi}{6} \). The third quadrant is the section of the unit circle where angles measure between \(180^{\circ} \) and \(270^{\circ} \), or in radians, between \(\pi \) and \(\frac{3\pi}{2} \).

• In the third quadrant, the cosine and sine values of angles are both negative because both the x and y coordinates of points in this quadrant are negative.

Knowing where an angle is located allows you to predict not only the sign of trigonometric functions but also helps in identifying the reference angle. With this, it becomes straightforward to apply other trigonometric properties and characteristics, such as the even nature of cosine, to find the exact value of the function.