Problem 16 - Precalculus Mathematics for Calculus

Question

Find the exact value of the trigonometric function at the given real number. (a) \(\cos \frac{7 \pi}{6}\) (b) \(\sec \frac{7 \pi}{6}\) (c) \(\csc \frac{7 \pi}{6}\)

Step-by-Step Solution

Answer

(a) -\(\frac{\sqrt{3}}{2}\), (b) -\(\frac{2\sqrt{3}}{3}\), (c) -2

Step 1: Convert to a reference angle

We start by converting the angle \(\frac{7\pi}{6}\) to its reference angle. First, we note that \(\frac{7\pi}{6}\) is in the third quadrant of the unit circle. The reference angle is computed by subtracting \(\pi\) from \(\frac{7\pi}{6}\): \(\frac{7\pi}{6} - \pi = \frac{7\pi}{6} - \frac{6\pi}{6} = \frac{\pi}{6}\). So, the reference angle is \(\frac{\pi}{6}\).

Step 2: Find cosine of the reference angle

The cosine of the reference angle \(\frac{\pi}{6}\) is \(\cos \frac{\pi}{6} = \frac{\sqrt{3}}{2}\). However, since \(\frac{7\pi}{6}\) is in the third quadrant where cosine is negative, we have \(\cos \frac{7\pi}{6} = -\frac{\sqrt{3}}{2}\).

Step 3: Find secant from cosine value

Since secant is the reciprocal of cosine, we find that \(\sec \frac{7\pi}{6} = \frac{1}{\cos \frac{7\pi}{6}} = \frac{1}{-\frac{\sqrt{3}}{2}} = -\frac{2}{\sqrt{3}}\). Rationalizing the denominator, we multiply the numerator and the denominator by \(\sqrt{3}\), giving us \(-\frac{2\sqrt{3}}{3}\).

Step 4: Find sine of the reference angle

The sine of \(\frac{\pi}{6}\) is \(\sin \frac{\pi}{6} = \frac{1}{2}\). Since \(\frac{7\pi}{6}\) is in the third quadrant where sine is also negative, we have \(\sin \frac{7\pi}{6} = -\frac{1}{2}\).

Step 5: Find cosecant from sine value

Cosecant is the reciprocal of sine, thus \(\csc \frac{7\pi}{6} = \frac{1}{\sin \frac{7\pi}{6}} = \frac{1}{-\frac{1}{2}} = -2\).

Key Concepts

Reference Angle

When dealing with trigonometric functions, the reference angle is a commonly used concept that helps simplify calculations. A 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, typically between 0 and \( \frac{\pi}{2} \) radians or 0 and 90 degrees.
To find the reference angle, it's important to first determine which quadrant the angle lies in. Here, the angle \( \frac{7\pi}{6} \) falls in the third quadrant. In this case, the reference angle can be found by subtracting \( \pi \) from the given angle:

• \( \frac{7\pi}{6} - \pi = \frac{\pi}{6} \)

This retains the angle within a manageable range, making it easier to use standard trigonometric values associated with \( \frac{\pi}{6} \). By understanding and applying the concept of reference angles, you’ll gain precision and make your calculations more straightforward.

Unit Circle

The unit circle is a powerful tool in trigonometry, representing all the points where the radius equals 1. Each point on the unit circle corresponds to an angle, helping visualize trigonometric function values. For any angle \( \theta \) in the unit circle, \( ( \cos(\theta), \sin(\theta) ) \) are the coordinates.
In our example, \( \frac{7\pi}{6} \) is positioned in the third quadrant of the unit circle. Here, both sine and cosine values are negative.

• The cosine of \( \frac{7\pi}{6} \) is \( -\frac{\sqrt{3}}{2} \).
• The sine of \( \frac{7\pi}{6} \) is \( -\frac{1}{2} \).

Using the unit circle means you can seamlessly derive trigonometrical identities and relationships, like reciprocal functions. Through practice, you'll see how the unit circle provides a visual method to understanding the core behavior of trigonometric functions across various quadrants.

Reciprocal Trigonometric Functions

Reciprocal trigonometric functions are another essential aspect of trigonometry, offering further insights into the relationships between angles and their trigonometric values. The primary reciprocal functions include secant (\( \sec(\theta) \)), cosecant (\( \csc(\theta) \)), and cotangent (\( \cot(\theta) \)).
Reciprocal functions are derived from the basic trigonometric functions:

• Secant is the reciprocal of cosine: \(\sec(\theta) = \frac{1}{\cos(\theta)}\).
  Thus, \(\sec \frac{7\pi}{6} = -\frac{2\sqrt{3}}{3}\).
• Cosecant is the reciprocal of sine: \(\csc(\theta) = \frac{1}{\sin(\theta)}\).
  Therefore, \(\csc \frac{7\pi}{6} = -2\).

Understanding these reciprocal relationships allows you to further interpret trigonometric equations and solve problems involving angles and their proportional sides. With this knowledge, you can connect the dots across different parts of trigonometry and solve complex problems with confidence.