Problem 17 - Algebra and Trigonometry

Question

Find the exact value of each expression. $$ \tan ^{-1}(-\sqrt{3}) $$

Step-by-Step Solution

Answer

\(-\pi/3\)

Step 1: Understanding the Inverse Tangent Function

The inverse tangent function \(\tan^{-1}(x)\) is defined as the angle whose tangent is \(x\). So when we are asked for \(\tan^{-1}(-\sqrt{3})\), we are really being asked for the angle whose tangent is \(-\sqrt{3}\). Based on this, and knowing that tangent values are positive in the first and third quadrants, and negative in the second and fourth quadrants, we need to find the angle in the fourth quadrant where \(\tan(angle) = -\sqrt{3}\).

Step 2: Identifying the Right Angle

The value of tangent is \(\sqrt{3}\) at the angles \(\pi/3\) in the first quadrant and \(-\pi/3\) in the fourth quadrant respectively. However, since the problem asks for \(-\sqrt{3}\), the inverse tangent of \(-\sqrt{3}\) is in the fourth quadrant. Therefore, the angle we are looking for is \(-\pi/3\).

Step 3: Concluding The Answer

We have determined that the angle in the fourth quadrant where \(\tan(angle) = -\sqrt{3}\) is at \(-\pi/3\). Thus, the exact value of the expression \(\tan^{-1}(-\sqrt{3})\) is \(-\pi/3\).

Key Concepts

Tangent Function

The tangent function is one of the primary trigonometric functions and is defined as the ratio of the opposite side to the adjacent side in a right triangle. If you have an angle \(\theta\), the tangent is represented as:\[ \tan(\theta) = \frac{\text{opposite side}}{\text{adjacent side}} \]Tangent is particularly special because it repeats its values every \(\pi\), making it a periodic function. This periodic nature of tangent allows us to analyze it across different quadrants. The function has positive values in the first and third quadrants, while it is negative in the second and fourth quadrants. Inverse tangent, or \(\tan^{-1}(x)\), refers to the angle whose tangent is \(x\). When we compute \(\tan^{-1}(-\sqrt{3})\), we seek the angle where the tangent equals \(-\sqrt{3}\). Understanding these relationships helps in solving many trigonometric equations efficiently.

Remember, the inverse function will yield an angle that lies between \(-\frac{\pi}{2}\) and \(\frac{\pi}{2}\). This ensures we are typically dealing with angles in the first and fourth quadrants.

Quadrants in Trigonometry

The Cartesian plane is divided into four quadrants, each affecting the sign of trigonometric functions:

• **First Quadrant**: All trigonometric functions are positive.
• **Second Quadrant**: Sine is positive, but cosine and tangent are negative.
• **Third Quadrant**: Tangent is positive, while sine and cosine are negative.
• **Fourth Quadrant**: Cosine is positive, but sine and tangent are negative.

When it comes to interpreting inverse trigonometric functions, the quadrant matters a lot. For \(\tan^{-1}(x)\), the angle is typically taken from the fourth quadrant if \(x\) is negative.

In the problem, since we are working with \(-\sqrt{3}\), we look at the fourth quadrant where the angles range from \(0\) to \(-\frac{\pi}{2}\). Understanding which quadrant contains the solution is crucial for identifying the correct angle when dealing with trigonometric functions.

Exact Values of Trigonometric Functions

Exact values are critical when working with common angles like \(30^\circ\), \(45^\circ\), and \(60^\circ\). These angles are often expressed in radians: \(\pi/6\), \(\pi/4\), and \(\pi/3\), respectively.

• For \(\pi/3\), \(\tan(\pi/3) = \sqrt{3}\).
• For \(-\pi/3\), which is strategically located in the fourth quadrant, \(\tan(-\pi/3) = -\sqrt{3}\).

When asked to find the inverse tangent \(\tan^{-1}(-\sqrt{3})\), we determine it as \(-\pi/3\) based on these exact values. This exactitude matters, as approximations might lead to errors, especially in advanced studies or applications.

By committing these key values to memory, not only do we solve problems more efficiently, but we also build a robust foundational understanding valuable across various areas of mathematics.