Problem 9 - Calculus Single Variable

Question

Calculate the value of the given inverse trigonometric function at the given point. $$ \arctan (-\sqrt{3}) $$

Step-by-Step Solution

Answer

The value of \( \arctan(-\sqrt{3}) \) is \(-\pi/3\).

Step 1: Understanding the Problem

We are asked to find the value of the inverse tangent function, \( \arctan (-\sqrt{3}) \). This means we need to determine the angle \( \theta \) such that \( \tan \theta = -\sqrt{3} \). The angle should be in the principal range of the \( \arctan \) function, which is \( (-\pi/2, \pi/2) \).

Step 2: Identifying the Reference Angle

First, find the reference angle where \( \tan \theta = \sqrt{3} \). The basic angle that has a tangent of \( \sqrt{3} \) is \( \pi/3 \). However, since tangent is negative in our problem, we need to place \( \pi/3 \) in a quadrant where tangent is negative in the \( \arctan \) range.

Step 3: Finding the Correct Quadrant

In the range \((-\pi/2, \pi/2)\), tangent is negative in the fourth quadrant. The angle in the fourth quadrant that corresponds to \( \pi/3 \) in the first quadrant is \(-\pi/3 \).

Step 4: Validating the Result

Verify that \( \tan(-\pi/3) = -\sqrt{3} \). Indeed, tangent of \(-\pi/3\) is \(-\tan(\pi/3) = -\sqrt{3}\). Hence, \( \arctan (-\sqrt{3}) = -\pi/3 \) is confirmed.

Key Concepts

Arctan Function

The arctan function, short for 'inverse tangent', allows us to find the angle whose tangent is a given value. When you see \( \arctan(x) \), you're seeking an angle \( \theta \) such that \( \tan(\theta) = x \). This function is handy when you want to discover angles from known tangent values. It's crucial to know that the arctan function adheres to a specific range, ensuring each output is unique. This brings us to the "principal range", which is chosen because tangent values repeat every \( \pi \). The unique outputs of the arctan function fall between \(-\frac{\pi}{2}\) and \(\frac{\pi}{2}\), ensuring our results aren't ambiguous. If the tangent of an angle is negative, the corresponding arctan value will typically be found in the fourth quadrant of this range. This is exactly why for \( \arctan(-\sqrt{3}) \), we end up with \(-\frac{\pi}{3}\).The arctan function is a fundamental part of trigonometry, helping students connect angles and their tangent values. Understanding it allows for solving a variety of mathematical problems.

Principal Range

When working with inverse trigonometric functions, the concept of the principal range is critical. For \( \arctan(x) \), the principal range refers to the angle output between \(-\frac{\pi}{2}\) and \(\frac{\pi}{2}\). This ensures that each result from the arctan function is unique and interprets appropriately within one full cycle of the tangent function.It's important because:

• It guarantees one-to-one correspondence, meaning every tangent has exactly one corresponding arctan value in this range.
• It avoids multiple interpretations of the same tangent value, which would otherwise occur in a periodic function like tangent.

The choice of \(-\frac{\pi}{2} < \theta < \frac{\pi}{2}\) is deliberate. In this range, the tangent function increases continuously, covering all possible real numbers. This aspect of the arctan function's range is why for a negative tangent (-\(\sqrt{3}\) here), the angle is located in the fourth quadrant, resulting in a negative angle such as \(-\frac{\pi}{3}\). Being mindful of these ranges helps in solving and verifying the results of trigonometric problems effectively.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are universally true for any angle. They serve as valuable tools in simplifying expressions and solving trigonometric equations. Key identities include:

• Pythagorean Identity: \( \sin^2(x) + \cos^2(x) = 1 \)
• Angle Sum Identities: \( \sin(a + b) = \sin a \cos b + \cos a \sin b \)
• Double Angle Formulas: \( \sin(2x) = 2 \sin x \cos x \)

In our example of \( \arctan(-\sqrt{3}) \), understanding trigonometric identities helps us make connections between the tangent function and desired angles. These identities support the calculation and verification processes. For instance, knowing that \( \tan(\pi/3) = \sqrt{3} \) helps identify reference angles efficiently.Using trigonometric identities, we can skillfully navigate from known values to unknown angles, helping to verify solutions such as \( \arctan(-\sqrt{3}) = -\frac{\pi}{3} \). Recognizing and utilizing these properties eases the task of working through inverse trigonometric problems.