Problem 6 - Algebra and Trigonometry

Question

Find the exact value of each expression, if it is defined. (a) \(\tan ^{-1} 1\) (b) \(\tan ^{-1}(-1)\) (c) \(\tan ^{-1} 0\)

Step-by-Step Solution

Answer

(a) \( \frac{\pi}{4} \), (b) \(-\frac{\pi}{4} \), (c) 0.

Step 1: Understanding Inverse Tangent Function

The inverse tangent function, \( \tan^{-1} \, x \), also known as arctangent, is used to find the angle \( \theta \) such that \( \tan(\theta) = x \). The angle \( \theta \) is usually in the range \(-\frac{\pi}{2} \) to \( \frac{\pi}{2} \).

Step 2: Solve \( \tan^{-1} 1 \)

The expression \( \tan^{-1} 1 \) asks for an angle \( \theta \) such that \( \tan(\theta) = 1 \). One such angle within the range \(-\frac{\pi}{2} \) to \( \frac{\pi}{2} \) is \( \frac{\pi}{4} \) because \( \tan\left( \frac{\pi}{4} \right) = 1 \). Thus, \( \tan^{-1} 1 = \frac{\pi}{4} \).

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

For \( \tan^{-1}(-1) \), we seek an angle \( \theta \) such that \( \tan(\theta) = -1 \). Within the range, \( \theta = -\frac{\pi}{4} \) satisfies this since \( \tan\left(-\frac{\pi}{4}\right) = -1 \). Thus, \( \tan^{-1}(-1) = -\frac{\pi}{4} \).

Step 4: Solve \( \tan^{-1} 0 \)

The expression \( \tan^{-1} 0 \) requires an angle \( \theta \) such that \( \tan(\theta) = 0 \). The angle \( \theta = 0 \) satisfies this condition since \( \tan(0) = 0 \). Therefore, \( \tan^{-1} 0 = 0 \).

Key Concepts

Arctangent

The arctangent, often denoted as \( \tan^{-1} x \) or \( \text{atan}(x) \), is one of the inverse trigonometric functions concentrated on reversing the tangent operation. Unlike the regular tangent function, which takes an angle and returns a ratio, the arctangent goes the other way. It takes a ratio and returns an angle. This angle is explicitly confined to a range between \( -\frac{\pi}{2} \) and \( \frac{\pi}{2} \). This restriction is crucial as it ensures each input will have a single output, making \( \tan^{-1} x \) a valid function.
When given an expression like \( \tan^{-1} 1 \), we're asked to determine the angle \( \theta \) where the tangent of that angle equals 1, lying within the specified range. The result \( \theta = \frac{\pi}{4} \) is a typical output for such calculations since \( \tan\left( \frac{\pi}{4} \right) = 1 \). Similarly, \( \tan^{-1}(-1) \) results in \( \theta = -\frac{\pi}{4} \), and \( \tan^{-1} 0 \) yields \( \theta = 0 \). These answers all fall within the allowable range that upholds the function's definitiveness.

Trigonometric Functions

Trigonometric functions are fundamental tools in mathematics, especially when dealing with angles and periodic phenomena. They include sine, cosine, tangent, and their inverses.
The inverse functions, such as arctangent, are crucial because they allow us to find angles when given the value of these trigonometric ratios. Here are some key points:

• Inverse Tangent: Since \( \tan(\theta) \) gives us the ratio of the opposite to adjacent sides in a right triangle, the inverse tangent tells us the angle \( \theta \) corresponding to such a ratio.
• Domains and Ranges: Each trigonometric function covers a range of angles, while its inverse dictates a corresponding range of results. For example, tangent's inverse arctangent range is \(-\frac{\pi}{2} \leq \theta \leq \frac{\pi}{2} \).
• Periodicity and Patterns: While the arctangent is limited in its range to maintain function integrity, the tangent function itself is periodic, repeating every \( \pi \) radians.

This periodicity of tangent explains the multiple equivalent angles for functions outside the principal range of its inverse.

Angle Measurement

Angle measurement is a significant aspect of trigonometry, focusing on determining the size of an angle in degrees or radians. Radians are commonly used in mathematics because they are a natural way of measuring angles, corresponding directly to the arc length on a unit circle.

• Radians: Angles are measured in radians when expressed as a ratio of the arc length to the radius of a circle. A full circle contains \( 2\pi \) radians.
• Degrees: Another unit for measuring angles is based on dividing the circle into 360 parts. Sometimes less intuitive in pure mathematics, but often used in practical contexts.
• Conversion: To convert between degrees and radians, use the relation \( \pi \) radians = 180 degrees. Therefore, \( \frac{\pi}{4} \) radians is equivalent to 45 degrees, which is often handy to check your answers in exercises involving arctangent.

Comprehending angle measurement helps grasp the answers produced by inverse trigonometric functions and understand the context of those calculations in more practical terms.