Problem 11 - A Graphical Approach to Precalculus with Limits

Question

Use the even-odd identities to write of the following expressions as a trigonometric function of a positive number $$\tan \left(-\frac{\pi}{7}\right)$$

Step-by-Step Solution

Answer

\(-\tan\left(\frac{\pi}{7}\right)\) is the expression as a trigonometric function of a positive number.

Step 1: Identify Even-Odd Identities

Recall that the tangent function is an odd function. This means that for any angle \( \theta \), \( \tan(-\theta) = -\tan(\theta) \). We'll use this property to rewrite the given expression.

Step 2: Apply the Odd Property of Tangent

Applying the odd function property, we have \( \tan\left(-\frac{\pi}{7}\right) = -\tan\left(\frac{\pi}{7}\right) \).

Step 3: Express as Positive Angle

The expression \( -\tan\left(\frac{\pi}{7}\right) \) is now a trigonometric function of a positive angle \( \frac{\pi}{7} \).

Key Concepts

even-odd identities

In trigonometry, even-odd identities are crucial rules that help simplify expressions involving trigonometric functions. These identities are based on whether a function is classified as \'even\' or \'odd\'. Functions are termed \'even\' if they satisfy the condition \( f(-x) = f(x) \), and \'odd\' if they satisfy \( f(-x) = -f(x) \). The even-odd property makes it possible to convert expressions of negative angles into expressions with positive angles.

For example, cosine \(\cos(x)\) is an even function because \(\cos(-x) = \cos(x)\). In contrast, sine \(\sin(x)\) and tangent \(\tan(x)\) are odd functions, which means \(\sin(-x) = -\sin(x)\) and \(\tan(-x) = -\tan(x)\). Understanding these identities allows you to seamlessly flip negative angle expressions to positive ones, which can simplify solving many trigonometric problems.

• The cosine function is even: \(\cos(-x) = \cos(x)\)
• The sine function is odd: \(\sin(-x) = -\sin(x)\)
• The tangent function is odd: \(\tan(-x) = -\tan(x)\)

tangent function

The tangent function, denoted as \(\tan(x)\), is one of the primary trigonometric functions. It is defined as the ratio of the sine function to the cosine function, \(\tan(x) = \frac{\sin(x)}{\cos(x)}\). This function plays a crucial role in trigonometry as it relates angles in right triangles to ratios of two legs.

Knowing that the tangent function is an odd function helps in solving problems with negative angles. If you know the value for the positive angle, you can easily find the value for the negative angle by simply taking the negative of the positive angle result.

In the exercise, you started with \(\tan\left(-\frac{\pi}{7}\right)\). Through the property of the odd tangent function, it switched to \(-\tan\left(\frac{\pi}{7}\right)\).

• The tangent is undefined at \(\frac{\pi}{2}\), \(\frac{3\pi}{2}\), etc.
• It has a period of \(\pi\), meaning \(\tan(x+\pi) = \tan(x)\).
• The tangent function graphs into a series of repeated curves, where each period has asymptotes at every odd multiple of \(\frac{\pi}{2}\).

positive angle

Positive angles are crucial in trigonometry as they provide a standard approach for dealing with angles and make problem-solving more straightforward. A positive angle moves counterclockwise from the positive x-axis, while a negative angle proceeds clockwise.

In trigonometric problems, rephrasing negative angles as positive angles can lead to easier solutions and interpretations. With the help of even-odd identities, especially with odd functions like tangent, we can transform \(\tan(-\theta)\) into \(-\tan(\theta)\). This eliminates negative values in trigonometric expressions, simplifying the analysis and calculations involved.

For the original exercise, turning \(\tan\left(-\frac{\pi}{7}\right)\) into \(-\tan\left(\frac{\pi}{7}\right)\) made it possible to work with the positive angle \(\frac{\pi}{7}\). This positive transformation often leads to solutions that are easier to conceptualize and apply, especially when evaluating or graphing trigonometric functions. Changing to positive angles standardizes the approach and generally aligns with the conventional way of interpreting angle measurements.

• Positive angles are measured counterclockwise.
• Negative angles are measured clockwise.
• Transforming negative angles into positive ones helps maintain consistency in trigonometric calculations.