Problem 77 - A Graphical Approach to Precalculus with Limits

Question

Give the exact real number value of each expression. Do not use a calculator. $$\tan ^{-1}\left(\tan \frac{5 \pi}{6}\right)$$

Step-by-Step Solution

Answer

The exact value is \(-\frac{\pi}{6}\).

Step 1: Understanding the Inverse Tan Function

The function \( an^{-1}(x)\), also known as the arctangent, reverses the tangent function, and is defined on the interval \(-\frac{\pi}{2}, \frac{\pi}{2}\). It means that the angle found by \( an^{-1}\) should be within this principal interval.

Step 2: Evaluating the Tangent Function

The tangent of an angle is defined as the ratio of sine to cosine. Compute \(\tan \frac{5\pi}{6}\):\[\tan \frac{5\pi}{6} = \frac{\sin \frac{5\pi}{6}}{\cos \frac{5\pi}{6}}\]Since \(\frac{5\pi}{6}\) is in the second quadrant:\(\sin \frac{5\pi}{6} = \frac{1}{2}\) and \(\cos \frac{5\pi}{6} = -\frac{\sqrt{3}}{2}\), so:\[\tan \frac{5\pi}{6} = \frac{\frac{1}{2}}{-\frac{\sqrt{3}}{2}} = -\frac{1}{\sqrt{3}} = -\frac{\sqrt{3}}{3}\]

Step 3: Finding the Arctangent of the Result

We now apply \(\tan^{-1}\) to the value obtained from the previous step. The result is \(-\frac{\sqrt{3}}{3}\). We need to find an angle in the interval \(-\frac{\pi}{2}, \frac{\pi}{2}\) such that its tangent is \(-\frac{\sqrt{3}}{3}\).The angle with \(\tan = -\frac{\sqrt{3}}{3}\) is \(-\frac{\pi}{6}\).

Step 4: Final Evaluation

Since \(\tan^{-1}\) operates over the interval \(-\frac{\pi}{2}, \frac{\pi}{2}\), our result is valid. Therefore, \( an^{-1}\left(\tan \frac{5 \pi}{6}\right) = -\frac{\pi}{6}\).

Key Concepts

Arctangent

The arctangent, denoted as \( \tan^{-1}(x) \), is an inverse trigonometric function that helps us find an angle whose tangent is \( x \). It's important because it reverses the effect of the tangent function, which means if you know the tangent of an angle, you can use arctangent to figure out that angle itself.
The arctangent is defined over all real numbers, but its range, or set of output values, is restricted to the interval \( (-\frac{\pi}{2}, \frac{\pi}{2}) \). This interval is important because it helps us maintain consistency and avoid potential confusion with angles that have the same tangent value. For instance, knowing that \( \tan \left( -\frac{\pi}{6} \right) = -\frac{\sqrt{3}}{3} \) helps us find the angle quickly without exploring all possibilities. It's like having a reliable path through a maze of angles when seeking the correct one related to a specific tangent value.

• Think of arctangent as answering the question, "What angle has this given tangent value?"
• Key feature: It reassures us that angles are always within the chosen principal range, so we can consistently interpret results.

Tangent Function

The tangent function, \( \tan(\theta) \), is one of the basic trigonometric functions and is defined as the ratio of sine to cosine. That means for any angle \( \theta \), the tangent is calculated by dividing the sine of the angle by the cosine. So, \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \).
In trigonometry, the tangent function can take any real number, meaning it can stretch from \(-\infty\) to \(+\infty\). However, the function has undefined points where the cosine of \( \theta \) equals zero, leading to vertical asymptotes on its graph.
For angles like \( \frac{5\pi}{6} \), computing tangent involves inspecting the sine and cosine values of the angle. Since \( \frac{5\pi}{6} \) is in the second quadrant of the unit circle:

• \( \sin \frac{5\pi}{6} = \frac{1}{2} \)
• \( \cos \frac{5\pi}{6} = -\frac{\sqrt{3}}{2} \)
• From here, \( \tan \frac{5\pi}{6} = \frac{\frac{1}{2}}{-\frac{\sqrt{3}}{2}} = -\frac{\sqrt{3}}{3} \)

This explanation of the tangent function lays the foundation for understanding how we transition to inverse operations like finding the arctangent.

Principal Values

Principal values refer to the restricted output range for inverse trigonometric functions, helping us unambiguously determine the angle associated with a given value. For the arctangent function, the principal values range from \( -\frac{\pi}{2} \) to \( \frac{\pi}{2} \).
This concept is crucial because the tangent function is periodic, meaning it repeats its values at regular intervals. Without restrictions, the arctangent could point to multiple, incorrect angles for the same tangent value. By using principal values, we simplify the verification process, ensuring consistent results.
For example, if \( \tan(\theta) = x \), and you want to find \( \theta \) using \( \tan^{-1} \), you're assured that \( \theta \) will be within its principal range. So, when given a task like finding \( \tan^{-1}(\tan \frac{5\pi}{6}) \), the result must firmly reside within the principal interval, yielding \( -\frac{\pi}{6} \) instead, as it correctly fits within the accepted range.

• Principal values prevent confusion by narrowing down the possible results to a standard, interpretable range.
• They ensure that inverse operations return the simplest and most direct angle possible.