Question

Simplify the expression. \(\tan (x-2 \pi)\)

Short answer

\(\tan(x - 2 \pi) = \tan(x)\)

Step-by-step solution

Apply the Periodicity of the Tangent Function

The tangent function has a period of \( \pi \). This periodicity implies that for any angle \( x \) and any integer \( n \), \( \tan(x) = \tan(x + n \pi) \). Therefore, \( \tan(x - 2 \pi) = \tan(x) \).

Key concepts

Periodicity of Trigonometric Functions

When studying trigonometry, understanding the concept of periodicity is crucial. Periodicity refers to the attribute of a function to repeat its values at regular intervals, known as periods. In the trigonometric family, functions like sine, cosine, and tangent exhibit periodic behavior.

For instance, the period of the sine and cosine functions is \(2\pi\), meaning that after an interval of \(2\pi\), these functions will start to repeat their values. However, the tangent function, which we're specifically focusing on in this context, has a period of \(\pi\). Intuitively, this means \(\tan(x)\) will yield the same value as \(\tan(x+\pi)\) or \(\tan(x+2\pi)\), and so on for any integer multiple of \(\pi\). This property is essential for simplifying trigonometric expressions, as depicted in the given exercise where \(\tan(x-2\pi)\) simplifies to \(\tan(x)\).

Understanding this periodic nature allows us to solve trigonometric equations more effectively and can often simplify complex problems into more manageable ones.

Simplify Trigonometric Expressions

Trigonometric expressions can sometimes appear daunting due to their complex-looking nature. However, with a solid grasp of trigonometric identities and properties, simplifying these expressions becomes a straightforward task. Identities such as the Pythagorean identities, angle sum and difference identities, and the aforementioned periodicity can be incredibly helpful.

To simplify a trigonometric expression effectively, it is important to look for opportunities to apply these identities. For the tangent function, knowing that it has a period of \(\pi\) can be instrumental, just as seen with \(\tan(x-2\pi)\) reducing to \(\tan(x)\) due to the period being an integer multiple of the function's periodicity. It's similarly useful to recognize when an expression can be restated using a fundamental identity to make calculations simpler.

Properties of Tangent

The tangent function, represented as \(\tan(x)\), is one of the six primary trigonometric functions and has some unique properties that are worth understanding. Aside from its periodicity of \(\pi\) discussed earlier, the tangent function is odd, meaning that \(\tan(-x) = -\tan(x)\). This property allows us to often determine the sign of the tangent function's value based on the angle’s quadrant.

The function is also undefined at odd multiples of \(\frac{\pi}{2}\), which corresponds to angles where the cosine function is zero since \(\tan(x) = \frac{\sin(x)}{\cos(x)}\). This last property is extremely relevant when dealing with expressions involving tangent, as it guides us to know where the function will encounter asymptotes, which are vertical lines that the graph approaches but never touches. Moreover, recognizing where these properties apply can greatly aid in graphing, solving, and simplifying expressions involving tangent.