Problem 7 - Algebra and Trigonometry

Question

\(3-10=\) Find the exact value of each expression, if it is defined. $$ \begin{array}{llll}{\text { (a) } \tan ^{-1}(-1)} & {\text { (b) } \tan ^{-1} \sqrt{3}} & {} & {\text { (c) } \tan ^{-1} \frac{\sqrt{3}}{3}}\end{array} $$

Step-by-Step Solution

Answer

(a) \(-\frac{\pi}{4}\), (b) \(\frac{\pi}{3}\), (c) \(\frac{\pi}{6}\).

Step 1: Understanding the Problem

The exercise asks us to find the exact values of the given inverse tangent expressions: \(\tan^{-1}(-1)\), \(\tan^{-1}(\sqrt{3})\), and \(\tan^{-1}\left(\frac{\sqrt{3}}{3}\right)\). We need to determine which angle (in radians) will give these tangent values.

Step 2: Review Inverse Trig Functions

The function \(\tan^{-1}(x)\) returns the angle \(\theta\) whose tangent is \(x\). The output of \(\tan^{-1}(x)\) is an angle \(\theta\) in radians, usually within the range \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\).

Step 3: Solve Part (a) \(\tan^{-1}(-1)\)

We are looking for the angle \(\theta\) such that \(\tan(\theta) = -1\). The angle whose tangent is \(-1\) and lies within the range \(-\frac{\pi}{2}, \frac{\pi}{2}\) is \(-\frac{\pi}{4}\).

Step 4: Solve Part (b) \(\tan^{-1}(\sqrt{3})\)

We are looking for the angle \(\theta\) such that \(\tan(\theta) = \sqrt{3}\). In the interval from \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\), the angle with tangent \(\sqrt{3}\) is \(\frac{\pi}{3}\).

Step 5: Solve Part (c) \(\tan^{-1}\left(\frac{\sqrt{3}}{3}\right)\)

We need to find the angle \(\theta\) where \(\tan(\theta) = \frac{\sqrt{3}}{3}\). The angle satisfying this within the range \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\) is \(\frac{\pi}{6}\).

Step 6: Summarize the Results

Thus, the exact values are: (a) \(-\frac{\pi}{4}\), (b) \(\frac{\pi}{3}\), (c) \(\frac{\pi}{6}\).

Key Concepts

Arctangent

The arctangent function is an inverse trigonometric function denoted as \( \tan^{-1}(x) \). It finds the angle whose tangent is equal to the given number \( x \).

• The range of the arctangent function is between \(-\frac{\pi}{2}\) and \(\frac{\pi}{2}\), covering all angles in the first and fourth quadrants of the unit circle.
• The use of this range ensures that the function provides a single, unique value for each input \( x \).
• When calculating \( \tan^{-1}(x) \), you are essentially asking "which angle has a tangent of \( x \)?"

Knowing this range is essential, as it helps us determine the correct angle when solving for arctan values, such as \( \tan^{-1}(-1) = -\frac{\pi}{4} \). When working with inverse trigonometric functions, remember to always consider the specified interval of output.

Tangent Function

The tangent function, represented as \( \tan(\theta) \), is a fundamental trigonometric function that relates an angle \( \theta \) in a right-angled triangle to the ratio of the opposite side over the adjacent side.

• Mathematically, \( \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \).
• The function is periodic with a period of \( \pi \), meaning it repeats every \( \pi \) radians or \( 180^\circ \).
• Tangent is undefined for angles where the cosine of \( \theta \) equals zero, resulting in a vertical asymptote in its graph.

Understanding the function's properties enables you to solve problems involving angles with specific tangent values. For example, knowing that \( \tan(\frac{\pi}{4}) = 1 \) and \( \tan(-\frac{\pi}{4}) = -1 \) is crucial in determining inverse values like \( \tan^{-1}(-1) \). Regular practice with the unit circle and tangent values aids in mastering this topic.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for every angle in their domains. These identities help simplify expressions and solve trigonometric equations.

• A fundamental identity is the Pythagorean identity: \( \sin^2(\theta) + \cos^2(\theta) = 1 \).
• The tangent identity can be derived from the Pythagorean identity: \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \).
• Co-function identities, like \( \sin(\theta) = \cos(\frac{\pi}{2} - \theta) \), are useful in transformations and integrations.

Applying these identities correctly allows for the reduction of complex equations into solvable forms. For instance, using the identity relation ensures the correct interpretation of \( \tan^{-1}(\sqrt{3}) \), by identifying that \( \tan(\pi/3) = \sqrt{3} \), we can ascertain that \( \tan^{-1}(\sqrt{3}) = \frac{\pi}{3} \). Mastery of these identities greatly enhances your trigonometry skills.