Problem 16 - Algebra and Trigonometry

Question

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

Convert the given angle \( \frac{7\pi}{6} \) into degrees to better understand its position on the unit circle. \( \frac{7\pi}{6} \times \frac{180^\circ}{\pi} = 210^\circ \). The angle 210° is in the third quadrant.

Step 2: Evaluate \(\cos\frac{7\pi}{6}\)

In the third quadrant, cosine is negative. Reference angle for \(210^\circ\) is \(30^\circ\). Therefore, \(\cos(210^\circ) = -\cos(30^\circ) = -\frac{\sqrt{3}}{2}\).

Step 3: Evaluate \(\sec\frac{7\pi}{6}\)

The secant function is the reciprocal of the cosine function. Therefore, \(\sec\frac{7\pi}{6} = \frac{1}{\cos\frac{7\pi}{6}} = \frac{1}{-\frac{\sqrt{3}}{2}} = -\frac{2}{\sqrt{3}}\). Rationalize the denominator to get \(-\frac{2\sqrt{3}}{3}\).

Step 4: Evaluate \(\csc\frac{7\pi}{6}\)

The cosecant function is the reciprocal of the sine function. Since in the third quadrant sine is negative and \(\sin(210^\circ) = -\sin(30^\circ)\), therefore \(\sin(210^\circ) = -\frac{1}{2}\). Thus, \(\csc\frac{7\pi}{6} = \frac{1}{\sin\frac{7\pi}{6}} = \frac{1}{-\frac{1}{2}} = -2\).

Key Concepts

Exploring the Unit Circle

The unit circle is an essential tool in trigonometry, representing all angles and their associated trigonometric function values. It is a circle with a radius of one unit, centered at the origin of a coordinate plane. By using the unit circle, you can easily visualize the position and trigonometric function values of various angles.

The angles on the unit circle are usually expressed in radians, owing to their easy relation with arc lengths. The standard position of an angle is along the positive x-axis, and it progresses counter-clockwise. For example, the angle \( \frac{7\pi}{6} \) radiance equates to \( 210^\circ \) degrees. This angle falls in the third quadrant of the unit circle.

In the third quadrant, both the sine and cosine values are negative. Consequently, trigonometric functions like tangent, which depend on the ratio of sine to cosine, are positive. The cosine of an angle, measured along the x-axis, corresponds to the x-coordinate of the point where the terminal side of the angle intersects the unit circle. Thus by understanding an angle's position on the unit circle, we can determine trigonometric function values effectively.

Understanding the Reference Angle

A reference angle is a simplified version of any non-acute angle that helps us calculate trigonometric function values more easily. It is the smallest angle that a given angle makes with the x-axis. Reference angles possess the same trigonometric function values as the original angles, except for a potential change in sign according to the angle's quadrant.

To find the reference angle of any angle in standard position, measure the smallest angle to the closest x-axis. For example, the reference angle of \( 210^\circ \) is \( 30^\circ \). This particular angle is significant because its trigonometric functions are memorized easily:

• \( \sin(30^\circ) = \frac{1}{2} \)
• \( \cos(30^\circ) = \frac{\sqrt{3}}{2} \)

In the context of quadrant determination, if an angle like \( 210^\circ \) is in the third quadrant, you know that both sine and cosine values are negative. However, knowing the reference angle, you can quickly compute \( \cos(210^\circ) = -\cos(30^\circ) = -\frac{\sqrt{3}}{2} \). Reference angles offer a simplified viewpoint for complicated trigonometric calculations.

Delving into Reciprocal Identities

Reciprocal identities in trigonometry involve functions that are reciprocals of one another. These include secant (\(\sec\)), cosecant (\(\csc\)), and cotangent (\(\cot\)). They are important because they provide a different method for evaluating function values and are integral in many trigonometric equations.

Each reciprocal identity emerges from a primary trigonometric function:

• \( \sec \theta = \frac{1}{\cos \theta} \)
• \( \csc \theta = \frac{1}{\sin \theta} \)
• \( \cot \theta = \frac{1}{\tan \theta} \)

The step-by-step solutions often employ these identities for dealing with unfamiliar trigonometric expressions. For example, finding \( \sec \frac{7\pi}{6} \) based on \( \cos \frac{7\pi}{6} = -\frac{\sqrt{3}}{2} \) gives: \( \sec \frac{7\pi}{6} = \frac{1}{-\frac{\sqrt{3}}{2}} = -\frac{2\sqrt{3}}{3} \) after rationalizing the denominator. Understanding these reciprocal relationships is crucial for mastering trigonometry, as it enhances problem-solving strategies in various mathematical applications.