Problem 44 - Calculus Single Variable

Question

Use one or more of the basic trigonometric identities to derive the given identity. \(\tan (\theta+\pi)=\tan (\theta)\)

Step-by-Step Solution

Answer

\( \tan(\theta + \pi) = \tan(\theta) \) is verified using trigonometric identities.

Step 1: Recall the Addition Formula for Tangent

The formula for tangent of the sum of two angles is given by: \( \tan(a + b) = \frac{\tan(a) + \tan(b)}{1 - \tan(a)\tan(b)} \). We will use this formula to evaluate \( \tan(\theta + \pi) \).

Step 2: Apply the Identity for \( \pi \)

We know that \( \tan(\pi) = 0 \) because the tangent of \( \pi \) radians (or 180 degrees) is 0. We can substitute \( \pi \) in the addition formula: \( \tan(\theta + \pi) = \frac{\tan(\theta) + \tan(\pi)}{1 - \tan(\theta)\tan(\pi)} \).

Step 3: Simplify the Expression

Substitute \( \tan(\pi) = 0 \) into the expression: \( \tan(\theta + \pi) = \frac{\tan(\theta) + 0}{1 - \tan(\theta)\cdot 0} = \frac{\tan(\theta)}{1} = \tan(\theta) \). Thus, \( \tan(\theta + \pi) = \tan(\theta) \).

Key Concepts

Addition Formula for Tangent

The Addition Formula for Tangent is a handy tool for working with tangent in the realm of trigonometry. This formula allows you to find the tangent of the sum of two angles. It's expressed succinctly as:

• \( \tan(a + b) = \frac{\tan(a) + \tan(b)}{1 - \tan(a)\tan(b)} \)

This formula is crucial for problems where angles are added together, like when deriving or simplifying identities involving the tangent function. It shows how the tangent of a sum can be expressed in terms of the tangents of the individual angles themselves, offering a neat bridge between seemingly complex angles.
The beauty of this formula lies in its symmetry and its similarity to formulas for other trigonometric functions, such as sine and cosine. Applying this formula requires knowledge of the tangents of each individual angle and some algebraic manipulation to arrive at your answer. It helps you break down complex trigonometric expressions into manageable parts.

Tangent Function

The Tangent Function is one of the primary trigonometric functions, often abbreviated as \( \tan \). It's defined in the context of a right triangle as the ratio of the opposite side to the adjacent side. Mathematically, it's represented by:

• \( \tan(\theta) = \frac{\text{opposite side}}{\text{adjacent side}} \)

This function is periodic, with a period of \( \pi \) radians, meaning it repeats its values every 180 degrees. This periodicity showcases how the function behaves in cycles, offering predictions about its values at specific angles.
When dealing with the unit circle, the tangent of an angle corresponds to the length of the line segment that is tangent to the circle at a given point, intersecting the x-axis. Unlike sine and cosine, which are limited to values between -1 and 1, the tangent function can take on any real value, leading to its graphs having a distinctive set of asymptotes. These asymptotes occur because the tangent function tends towards infinity as it approaches certain angles.

Angle Addition in Trigonometry

Angle Addition in Trigonometry is a method used to handle problems involving the sum of two angles. It involves using formulas that relate the trigonometric functions of the summed angle to the trigonometric functions of the individual angles.
For trigonometric functions, such as tangent, these addition formulas are tools that break down more complex angles into simpler components, facilitating easier manipulation and solution. The addition formula for tangent, mentioned earlier, is a prime example of such a formula.
Understanding angle addition is imperative for solving trigonometric equations and identities efficiently. It not only simplifies computations but also helps in verifying the equivalence of trigonometric expressions, thereby playing an indispensable role in higher-level mathematics as well.

• These formulas aid in solving problems involving rotation, oscillation, and anything periodic, like sound waves or electrical currents. They provide the framework for understanding how different angles interact with each other within geometric contexts.