Problem 11 - Precalculus: Functions and Graphs

Question

Find the exact value. (a) \(\tan (5 \pi / 6)\) (b) \(\tan (-\pi / 3)\)

Step-by-Step Solution

Answer

(a) \(-\frac{1}{\sqrt{3}}\), (b) \(-\sqrt{3}\)

Step 1: Understand the Problem

We need to find the exact values of two trigonometric functions: \( \tan \left( \frac{5\pi}{6} \right) \) and \( \tan \left( -\frac{\pi}{3} \right) \). Use the properties of tangent and reference angles to solve them.

Step 2: Use the Unit Circle (for \( \tan \left( \frac{5\pi}{6} \right) \))

Determine where \( \frac{5\pi}{6} \) is located on the unit circle. It corresponds to 150° (convert radians to degrees: \( \frac{5\pi}{6} \times \frac{180}{\pi} = 150° \)). The reference angle is 30°.

Step 3: Find Tangent Values (for \( \tan \left( \frac{5\pi}{6} \right) \))

The reference angle is 30°, where \( \tan(30°) = \frac{1}{\sqrt{3}} \). However, \( \frac{5\pi}{6} \) is in the second quadrant, where tangent is negative. Therefore, \( \tan \left( \frac{5\pi}{6} \right) = -\frac{1}{\sqrt{3}} \).

Step 4: Use the Unit Circle (for \( \tan \left( -\frac{\pi}{3} \right) \))

Determine where \( -\frac{\pi}{3} \) is located. This is \(-60°\), corresponding to a clockwise rotation from the positive \(x\)-axis. Its reference angle is \(60°\) or \(\frac{\pi}{3}\).

Step 5: Find Tangent Values (for \( \tan \left( -\frac{\pi}{3} \right) \))

The tangent of 60° is \( \sqrt{3} \). Since tangent is negative in the fourth quadrant, where \(-60°\) lies, \( \tan \left( -\frac{\pi}{3} \right) = -\sqrt{3} \).

Key Concepts

Unit Circle

The unit circle is a fundamental concept in trigonometry, representing all the angles and their corresponding trigonometric function values in a circle with a radius of 1. When working with angles in trigonometric problems, it's essential to visualize or use the unit circle.
The circle is divided into degrees or radians, where the entire circle is 360° or \( 2\pi \) radians. Each point on the unit circle corresponds to a coordinate \((\cos \theta, \sin \theta)\), which helps us find sine and cosine values of an angle \( \theta \). Because the radius is 1, these coordinates also provide the exact trigonometric ratios.

• The positive x-axis is the starting point at 0° (or 0 radians).
• Counterclockwise movement describes positive angles, while clockwise movement describes negative angles.

The same principles apply to tangent, which is calculated using the ratio \( \tan \theta = \frac{\sin \theta}{\cos \theta} \). Knowing the positions on the unit circle helps in determining the signs and values of these trigonometric functions.

Reference Angles

Reference angles serve as crucial aids in simplifying trigonometric calculations. A reference angle is the acute angle that a given angle makes with the x-axis. It is usually denoted by the Greek letter \( \alpha \). Reference angles help us utilize known values from the first quadrant for calculating trigonometric functions in other quadrants.
To find a reference angle:

• For an angle in the first quadrant, the reference angle is the angle itself.
• For the second quadrant, subtract the angle from \( 180° \) or \( \pi \).
• For the third quadrant, subtract \( 180° \) or \( \pi \) from the angle.
• For the fourth quadrant, subtract the angle from \( 360° \) or \( 2\pi \).

In the case of \( \tan \left( \frac{5\pi}{6} \right) \), the reference angle is 30° or \( \frac{\pi}{6} \) radians. This helps us use the known value \( \tan(30°) = \frac{1}{\sqrt{3}} \) to find \( \tan \left( \frac{5\pi}{6} \right) \).

Tangent Function

The tangent function \( \tan \theta \) is one of the fundamental trigonometric functions, defined as the ratio of sine to cosine, or \( \tan \theta = \frac{\sin \theta}{\cos \theta} \).
Key points about the tangent function include:

• Tangent values grow indefinitely positive or negative as they approach angles where cosine equals zero, leading to vertical asymptotes.
• The function is periodic with a period of \( \pi \) radians or 180°.
• Because the tangent depends on both sine and cosine, its values heavily rely on the quadrant in which the angle lies.

For angles like \( \frac{5\pi}{6} \) and \( -\frac{\pi}{3} \), understanding the tangent function involves combining the reference angle's tangent value with the sign determined by the quadrant, giving us \( -\frac{1}{\sqrt{3}} \) and \( -\sqrt{3} \) respectively.

Quadrants

In trigonometry, the coordinate plane is divided into four quadrants by the x-axis and y-axis. Each quadrant holds specific characteristics that affect the values and signs of trigonometric functions:

• First Quadrant (0° to 90° or 0 to \( \frac{\pi}{2} \)): All trigonometric function values are positive.
• Second Quadrant (90° to 180° or \( \frac{\pi}{2} \) to \( \pi \)): Sine values are positive, while cosine and tangent are negative.
• Third Quadrant (180° to 270° or \( \pi \) to \( \frac{3\pi}{2} \)): Tangent is positive, while sine and cosine are negative.
• Fourth Quadrant (270° to 360° or \( \frac{3\pi}{2} \) to \( 2\pi \)): Cosine is positive, while sine and tangent are negative.

When dealing with angles like \( \frac{5\pi}{6} \) and \(-\frac{\pi}{3}\), it's crucial to identify which quadrant the angle resides in. This allows understanding how the sign of the tangent value is determined. For \( \frac{5\pi}{6} \), in the second quadrant, tangent values are negative; for \(-\frac{\pi}{3}\), in the fourth quadrant, tangent values are also negative.