Problem 42 - Precalculus Essentials

Question

Find the exact value of each expression, if possible. Do not use a calculator. $$\tan ^{-1}\left(\tan \frac{3 \pi}{4}\right)$$

Step-by-Step Solution

Answer

The value of the expression is \(-\frac{π}{4}\).

Step 1: Understanding the inverse tangent function

The inverse tangent function, denoted as \(\tan^{-1}(x)\), is the inverse function of the tangent function. It returns the angle whose tangent is \(x\). It's important to note that \(\tan^{-1}(\tan(x)) = x\) only if \(x\) is within the interval \(-\frac{π}{2} \leq x \leq \frac{π}{2}\).

Step 2: Evaluate the tangent function

Evaluate the tangent function at \(\frac{3 \pi}{4}\) to get \(\tan(\frac{3 \pi}{4}) = -1\). -1 falls within the range of the tangent function, so we can find the inverse tangent of -1.

Step 3: Find the inverse tangent

The inverse tangent of -1, \(\tan^{-1}(-1)\), is \(-\frac{π}{4}\), which is a common value known from the unit circle.

Step 4: Final Solution

Thus, \(\tan^{-1}(\tan(\frac{3 \pi}{4}))=-\frac{π}{4}\).

Key Concepts

Tangent Function

The tangent function, often noted as \( \tan(x) \), relates to angles in right-angled triangles. It is the ratio of the opposite side to the adjacent side. This trigonometric function is periodic with a period of \( \pi \), meaning \( \tan(x) = \tan(x + \pi n) \) for any integer \( n \). The function repeats its values every \( \pi \) units. In the context of the unit circle, the tangent function is defined based on the coordinates of the point where the terminal side of the angle intersects the unit circle. It's crucial to remember that the tangent function has a range of all real numbers and an infinite domain; however, it has vertical asymptotes where the cosine is zero.

Additionally, the inverse tangent function, \( \tan^{-1}(x) \), finds the angle whose tangent is \( x \). Its principal range is \( -\frac{\pi}{2} \leq y \leq \frac{\pi}{2} \), making sure the function is one-to-one. This principal range ensures each \( \tan^{-1}(x) \) value maps precisely to a specific angle.

Unit Circle

The unit circle is a circle with radius 1 centered at the origin of a coordinate system. In trigonometry, it's an essential tool used to define sine, cosine, and tangent functions for all real numbers. By using this circle, every angle corresponds to a unique point on the circumference.

For a given angle \( \theta \), the point \((x, y)\) on the unit circle has coordinates where \( x = \cos(\theta) \) and \( y = \sin(\theta) \). For tangent, the function is represented as \( \tan(\theta) = \frac{y}{x} \).

The unit circle helps easily recognize common angle values, such as \( \frac{\pi}{4} \), \( \frac{3\pi}{4} \), and others, which are helpful when solving problems related to trigonometric functions without a calculator. The tangent of some specific angles, like \( \frac{3 \pi}{4} \), is known to be \(-1\), confirming the link between these standard angle measures and their trigonometric values.

Angle Conversion

Angle conversion is a method widely used in trigonometry to switch between degrees and radians. Understanding this conversion is vital, as it allows you to work seamlessly between different systems of angle measurement.

The key relationship to remember is that \( 180^\circ \) equals \( \pi \) radians. Therefore, to convert from radians to degrees, multiply by \( \frac{180}{\pi} \); to convert from degrees to radians, multiply by \( \frac{\pi}{180} \).

Commonly used angles in radians, such as \( \frac{3\pi}{4} \), are equivalent to specific degree measures. For instance, \( \frac{3\pi}{4} \) radians is equivalent to \( 135^\circ \). Mastering these conversions helps solve trigonometric problems efficiently, ensuring you can find angles in the desired format required by exercises, like identifying values on the unit circle or computing inverse trigonometric functions accurately.