Problem 26 - Algebra and Trigonometry

Question

9–32 Find the exact value of the trigonometric function. $$\tan \frac{5 \pi}{6}$$

Step-by-Step Solution

Answer

The exact value of \( \tan \frac{5 \pi}{6} \) is \(-\frac{\sqrt{3}}{3}\).

Step 1: Identify the Angle in Degrees

To convert the angle from radians to degrees, use the formula: \( \text{Degrees} = \frac{\pi}{180^{\circ}} \). For \( \frac{5\pi}{6} \), the conversion is \( \frac{5\pi}{6} \times \frac{180^{\circ}}{\pi} = 150^{\circ} \). Thus, \( \frac{5\pi}{6} \) radians is equivalent to \( 150^{\circ} \).

Step 2: Locate the Angle on the Unit Circle

The angle of \( 150^{\circ} \) is in the second quadrant. In this quadrant, the tangent of an angle is negative. The reference angle for \( 150^{\circ} \) is \( 180^{\circ} - 150^{\circ} = 30^{\circ} \).

Step 3: Determine the Tangent of the Reference Angle

The tangent of the reference angle \( 30^{\circ} \) is known to be \( \tan 30^{\circ} = \frac{1}{\sqrt{3}} \) or \( \frac{\sqrt{3}}{3} \) after rationalizing the denominator.

Step 4: Apply the Quadrant Rule

Since \( 150^{\circ} \) is in the second quadrant, where tangent values are negative, apply the sign to the tangent of the reference angle. Thus, \( \tan 150^{\circ} = -\frac{\sqrt{3}}{3} \).

Step 5: State the Exact Value

The exact value of \( \tan \frac{5 \pi}{6} \) is \(-\frac{\sqrt{3}}{3}\).

Key Concepts

Unit Circle

The unit circle is a fundamental tool in trigonometry. It is a circle with a radius of one unit, centered at the origin of a coordinate system. The unit circle helps us to define trigonometric functions such as sine, cosine, and tangent by using the coordinates of points on the circle.

When we talk about placing angles on the unit circle, we start from the positive x-axis and move counterclockwise for positive angles. This movement traces out an angle in standard position. Each point on the unit circle corresponding to an angle is \((\cos \theta, \sin \theta)\), which helps in finding trigonometric values. The tangent of an angle can be found using \(\tan \theta = \frac{\sin \theta}{\cos \theta}\).

• The unit circle allows understanding of angles beyond \(0^{\circ}\) to \(360^{\circ}\) or \(0\) to \(2\pi\) radians, by looping around the circle.
• It connects circular motion and trigonometric functions.

Angle Conversion

Converting angles between degrees and radians is a key skill in trigonometry. Angles in trigonometry can be expressed in both degrees and radians, where radians provide a more natural measure related to the circle.

To convert an angle from radians to degrees, you use the conversion factor\( \frac{180^{\circ}}{\pi} \), and for degrees to radians, use\( \frac{\pi}{180^{\circ}} \). For example, converting the angle\( \frac{5\pi}{6} \) radians to degrees involves multiplying by the ratio\( \frac{180^{\circ}}{\pi} \), resulting in \(150^{\circ}\).

• Conversion helps in understanding and using angles in different contexts.
• Both units have specific applications, radians are often used in calculus and mathematical analysis.

Tangent Function

The tangent function is a fundamental trigonometric function represented as\( \tan \theta \). It is defined as the ratio of the sine and cosine of an angle:\( \tan \theta = \frac{\sin \theta}{\cos \theta} \). Tangent is periodic, with a period of\( \pi \) (or \(180^{\circ}\)) which means it repeats its values every \(\pi\) radians.

The behavior of the tangent function varies in different quadrants:

• In the first quadrant, it is positive.
• In the second quadrant, it becomes negative.
• In the third quadrant, it is positive again.
• In the fourth quadrant, it turns negative once more.

For the angle\( \frac{5\pi}{6} \), which is \(150^{\circ}\), the tangent is negative because it is located in the second quadrant.

• The tangent function is undefined for angles where the cosine is zero, seen in vertical asymptotes at\( \frac{\pi}{2} + k\pi \).

Reference Angles

Reference angles are an essential concept when working on trigonometric functions across different quadrants. A reference angle is always the acute angle formed by the terminal side of the angle and the horizontal axis.

The reference angle helps simplify the process of finding trigonometric values for any given angle. It is calculated differently based on the quadrant:

• For angles in the first quadrant, the reference angle is the angle itself.
• In the second quadrant, it's \(180^{\circ} - \text{angle} \).
• For the third quadrant, use \(\text{angle} - 180^{\circ}\).
• In the fourth quadrant, it becomes \(360^{\circ} - \text{angle}\).

For \(150^{\circ}\), the reference angle is \(30^{\circ}\). Reference angles play a key role in determining the sign and value of trigonometric functions in various quadrants. They ensure that calculations remain consistent across the unit circle.