Problem 33 - Calculus Early Transcendentals: Pearson New International Edition

Question

Use the addition identity for the tangent to show that \(\tan (t+\pi)=\tan t\) for all \(t\) in the domain of \(\tan t\).

Step-by-Step Solution

Answer

\( \tan(t+\pi) = \tan t \) by using the tangent addition identity and \( \tan \pi = 0 \).

Step 1: Understand the Tangent Addition Identity

The tangent addition identity is given by \( \tan(a+b) = \frac{\tan a + \tan b}{1 - \tan a \tan b} \). We'll use this identity to express \( \tan(t+\pi) \).

Step 2: Substitute into Tangent Identity

Set \( a = t \) and \( b = \pi \) in the addition identity: \( \tan(t+\pi) = \frac{\tan t + \tan \pi}{1 - \tan t \tan \pi} \).

Step 3: Evaluate \( \tan \pi \)

Recall that \( \tan \pi = 0 \). Substitute this into the equation to get: \( \tan(t+\pi) = \frac{\tan t + 0}{1 - \tan t \cdot 0} = \frac{\tan t}{1} = \tan t \).

Step 4: Conclusion

Thus, we have shown that \( \tan(t+\pi) = \tan t \) using the tangent addition identity.

Key Concepts

Trigonometric Functions

Trigonometric functions are essential in mathematics, especially when dealing with angles and periodic phenomena. The primary trigonometric functions are sine, cosine, and tangent. These functions help to relate the angles of a triangle to its side lengths. They are also used in various fields, such as physics, engineering, and computer science.

Let's dive a bit deeper into each:

• Sine (\(\sin\)): This function relates the opposite side of a right triangle to its hypotenuse. The sine function is periodic with a period of \(2\pi\).
• Cosine (\(\cos\)): This function attaches the length of the adjacent side to the hypotenuse. Like sine, cosine has a period of \(2\pi\).
• Tangent (\(\tan\)): It is the ratio of the sine function to the cosine function (i.e., \(\tan \theta = \frac{\sin \theta}{\cos \theta}\)). This function is periodic with a period of \(\pi\).

These trigonometric functions allow us to solve various problems, including angular sums and the behavior of waves. Understanding these functions paves the way to grasp more advanced concepts like the tangent addition identity.

Tangent Function Property

The tangent function has unique properties that distinguish it from sine and cosine. One of these key properties is its periodicity, which means that it repeats its values at regular intervals. The periodicity of the tangent function is \(\pi\), which results from its definition as the ratio of sine and cosine.

Here are some important properties:

• The period of \(\tan \theta\) is \(\pi\).
• The tangent function is undefined at \(\theta = \frac{(2n+1)\pi}{2}\), where \(n\) is any integer, due to the cosine function being zero, causing a division by zero.
• Tangent's symmetry is around the origin, making it an odd function. This means \(\tan(-\theta) = -\tan(\theta)\).

These properties help explain why \(\tan(t+\pi) = \tan t\). When we encounter this equation, the periodicity plays a vital role as adding \(\pi\) merely cycles the function back to its starting point, thus not altering its value.

Mathematical Proof

When diving into mathematical proofs, the goal is to start with known equations or identities and transform them step-by-step into the desired result. In the case of the tangent addition identity, we use the formula:\[ \tan(a+b) = \frac{\tan a + \tan b}{1 - \tan a \tan b} \]

To prove the statement \(\tan(t+\pi) = \tan t\), we follow these logical steps:

• Step 1: Use the tangent addition identity and set \(a = t\) and \(b = \pi\).
• Step 2: Substitute into the identity: \(\tan(t+\pi) = \frac{\tan t + \tan \pi}{1 - \tan t\tan \pi}\)
• Step 3: Evaluate \(\tan \pi\), knowing the function has a value of zero at \(\pi\).
• Step 4: The expression simplifies as \(\tan(t+\pi) = \frac{\tan t}{1} = \tan t\).

Each calculation logically follows from the previous. It is this methodical approach, harnessing the properties of the tangent function, which facilitates understanding of why \(\tan(t+\pi) = \tan t\) is true for all \(t\). Such proofs not only verify trigonometric identities but also deepen our grasp of trigonometric functions and their behavior.