Problem 36 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 25-36: Verify the reduction formula. $$ \tan (x+\pi)=\tan x $$

Step-by-Step Solution

Answer

\( \tan(x + \pi) = \tan(x) \) verified by the periodicity of tangent.

Step 1: Understanding the Tangent Function

The tangent function, \( \tan(x) \), is defined as \( \tan(x) = \frac{\sin(x)}{\cos(x)} \). It is a trigonometric function that relates the angle of a right triangle to the ratio of the opposite side over the adjacent side.

Step 2: Exploring the Property of Tan

The property in question, \( \tan(x + \pi) = \tan(x) \), can be initially investigated by understanding that \( \pi \) is equivalent to 180 degrees; thus, adding \( \pi \) results in a shift across the x-axis on the unit circle.

Step 3: Applying the Periodicity of Tangent

Let's leverage the periodicity of the tangent function. The tangent function is periodic with a period of \( \pi \), meaning that adding \( \pi \) to its argument shifts the function by a full cycle, keeping the function values unchanged, i.e., \( \tan(x + \pi) = \tan(x) \).

Step 4: Verifying Using Trigonometric Identities

Recall the trigonometric identity: \( \tan(a + b) = \frac{\tan(a) + \tan(b)}{1 - \tan(a)\tan(b)}\). By substituting \( a = x \) and \( b = \pi \), and knowing \( \tan(\pi) = 0 \), we have: \( \tan(x + \pi) = \frac{\tan(x) + 0}{1 - \tan(x) \cdot 0} = \frac{\tan(x)}{1} = \tan(x) \).

Key Concepts

Tangent Function

The tangent function, denoted as \( \tan(x) \), plays a crucial role in trigonometry. It is fundamentally defined as the ratio of the sine and cosine functions: \( \tan(x) = \frac{\sin(x)}{\cos(x)} \). This means it relates an angle in a right triangle to the length of the opposite side divided by the length of the adjacent side.
This function is essential when working with angles and triangles, particularly in contexts like geometry, physics, and engineering. Unlike sine and cosine, the tangent function is undefined at certain angles, where \( \cos(x) = 0 \), because division by zero is undefined. These angles occur at intervals of \( \frac{\pi}{2} + n\pi \), where \( n \) is an integer.
The behavior of the tangent function can be visualized on the unit circle: as we move around the circle, the values for \( \tan(x) \) reflect the changing ratio of these two sides.

Periodicity

Periodicity is a fundamental characteristic of trigonometric functions, indicating that these functions repeat their values in regular intervals. For the tangent function, this interval is \( \pi \).

• This means that for any angle \( x \), \( \tan(x) = \tan(x + k\pi) \), where \( k \) is any integer.


• The function returns to its original value after each full cycle of \( \pi \), unlike sine and cosine which have a periodicity of \( 2\pi \).

The periodic nature of tangent makes it particularly useful for modeling periodic phenomena, even those outside traditional geometric contexts, such as oscillations or wave patterns. This property allows us to use the function over broad ranges of angles without changing the fundamental relationship described by the tangent function.

Unit Circle

The unit circle is a conceptual tool in trigonometry, providing a geometric perspective of trigonometric functions. It is a circle centered at the origin with a radius of one unit. All angles and trigonometric function values can be associated with points on this circle.
For tangent, specifically, it's beneficial to understand the unit circle because it helps visualize how \( \tan(x) \) changes as \( x \) increases. Moving through \( \pi \) radians on the unit circle corresponds to a half-circle turn, effectively mapping one period of the tangent function. On the unit circle:

• The tangent value at any angle corresponds to the length of a line tangent to the circle from the point on the circle at that angle, hence the name "tangent."


• When you add \( \pi \) to an angle \( x \), you shift to the diametrically opposite point, which explains why \( \tan(x + \pi) = \tan(x) \).

Reduction Formula

The reduction formula is a tool in trigonometry to simplify expressions by using properties of angles and trigonometric identities. In this context, the formula \( \tan(x + \pi) = \tan(x) \) is verified using these principles.
Here's how it works:

• The identity \( \tan(a + b) = \frac{\tan(a) + \tan(b)}{1 - \tan(a)\tan(b)} \) simplifies into \( \tan(x) \) because \( \tan(\pi) = 0 \), leaving us with \( \frac{\tan(x)}{1} = \tan(x) \).
• This formula uses \( b = \pi \), demonstrating that shifting an angle by this amount on the coordinate circle results in no change in the value of the tangent function.

This approach streamlines calculations and is especially useful in verifying identities and solving equations, providing a clear, concise method to account for the periodic behavior of the tangent function.