Problem 33 - 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. $$\tan \left(-\frac{11 \pi}{6}\right)$$

Step-by-Step Solution

Answer

The exact value of \( \tan \left(-\frac{11 \pi}{6}\right) \) is \(-\frac{\sqrt{3}}{3}\).

Step 1: Understand the properties of the function

The tangent function, \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \), depends on sine and cosine. Since sine is an odd function \( \sin(-\theta) = -\sin(\theta) \) and cosine is an even function \( \cos(-\theta) = \cos(\theta) \), it follows that \( \tan(-\theta) = -\tan(\theta) \).

Step 2: Determine the equivalent positive angle

Find the positive angle coterminal with \(-\frac{11 \pi}{6}\). Add \(2\pi\) (the full circle measure) to it: \(-\frac{11 \pi}{6} + 2\pi = -\frac{11 \pi}{6} + \frac{12 \pi}{6} = \frac{\pi}{6}\).

Step 3: Evaluate tangent at the positive angle

Now, calculate \( \tan\left(\frac{\pi}{6}\right) \). The unit circle provides that \( \sin\left(\frac{\pi}{6}\right) = \frac{1}{2} \) and \( \cos\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2} \). Therefore, \( \tan\left(\frac{\pi}{6}\right) = \frac{\sin\left(\frac{\pi}{6}\right)}{\cos\left(\frac{\pi}{6}\right)} = \frac{\frac{1}{2}}{\frac{\sqrt{3}}{2}} = \frac{1}{\sqrt{3}} \).

Step 4: Apply tangent identity for negative angles

Using \( \tan(-\theta) = -\tan(\theta) \), we find \( \tan\left(-\frac{11 \pi}{6}\right) = -\tan\left(\frac{\pi}{6}\right) = -\frac{1}{\sqrt{3}} \).

Step 5: Rationalize the denominator

To express \( -\frac{1}{\sqrt{3}} \) in a standard form, multiply the numerator and the denominator by \( \sqrt{3} \) to get \( -\frac{\sqrt{3}}{3} \).

Key Concepts

Unit Circle

The unit circle is a crucial tool for understanding trigonometry. It is a circle with a radius of 1 unit, centered at the origin of a coordinate plane. The circle allows us to connect angles with their respective trigonometric values. You can use the coordinates of points on the unit circle to determine sine and cosine values for different angles.

• Sine of an angle is the y-coordinate of its corresponding point on the unit circle.
• Cosine of an angle is the x-coordinate.
• Tangent is the ratio of the sine to the cosine, \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \).

Knowing these coordinates can help you find the exact values of trigonometric functions at various angles. For instance, at \( \frac{\pi}{6} \), the coordinates are \( \left(\frac{\sqrt{3}}{2}, \frac{1}{2}\right) \) which leads to the calculation of tangent as \( \tan\left(\frac{\pi}{6}\right) = \frac{1}{\sqrt{3}} \).

Tangent Function

The tangent function is derived from sine and cosine. It provides the slope of the line connecting the origin to a point on the unit circle. Essentially, the tangent of an angle is \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \). This ratio can interpret how steep or flat the line at an angle \( \theta \) is.Some key characteristics:

• The tangent function is periodic with period \( \pi \), meaning \( \tan(\theta + \pi) = \tan(\theta) \).
• The function is undefined where the cosine is zero, leading to vertical asymptotes.
• Common tangent values include \( \tan\left(\frac{\pi}{4}\right) = 1 \) and \( \tan\left(\frac{\pi}{6}\right) = \frac{1}{\sqrt{3}} \).

Understanding the tangent's relationship with sine and cosine helps determine its behavior over various angle ranges.

Odd and Even Functions

In trigonometry, recognizing odd and even functions can simplify complex calculations. Sine and cosine demonstrate these properties, influencing how the tangent function behaves.

• Odd Function: Sine is an odd function, which means \( \sin(-\theta) = -\sin(\theta) \). This symmetry implies that if you reverse the angle, the sine value also reverses.
• Even Function: Cosine is even, such that \( \cos(-\theta) = \cos(\theta) \). This signifies the cosine value remains the same, irrespective of the angle's direction.

These properties extend to the tangent function: \( \tan(-\theta) = -\tan(\theta) \). This tells us that tangent is also an odd function. It helps in calculations like \( \tan(-\frac{11 \pi}{6}) \) to know quickly that it's simply \( -\tan(\frac{11 \pi}{6}) \).

Angle Conversion

Angle conversion involves changing angles to different but equivalent measurements. This is often useful in dealing with negative angles and calculating trigonometric functions.To convert a negative angle to a positive one, we find the positive coterminal angle. A coterminal angle shares its terminal side with the given angle. The conversion formula is adding or subtracting \( 2\pi \) to shift the angle into a positive domain. For example, converting \( -\frac{11\pi}{6} \) to a positive angle involves:

• Adding \( 2\pi \) results in: \(-\frac{11\pi}{6} + \frac{12\pi}{6} = \frac{\pi}{6} \).
• This positive angle, \( \frac{\pi}{6} \), is easier to work with, especially when using the unit circle for calculations.

Understanding how to convert and find these angles is essential for precise computation in trigonometry.