Problem 43 - Algebra and Trigonometry

Question

\(23-44=\) Find the exact value of the expression, if it is defined. \(\sin \left(\tan ^{-1}(-1)\right)\)

Step-by-Step Solution

Answer

The value is \(-\frac{\sqrt{2}}{2}\).

Step 1: Understanding Inverse Trigonometric Functions

First, recall that \( \tan^{-1}(-1) \) represents the angle in radians whose tangent is -1. The tangent of an angle is the ratio of the sine to the cosine.

Step 2: Identify the Angle for \( \tan^{-1}(-1) \)

The angle we need satisfies \( \tan(\theta) = -1 \). In the range \(-\frac{\pi}{2} < \theta < \frac{\pi}{2}\) where \( \tan^{-1} \) is defined, \( \theta = -\frac{\pi}{4} \) because the tangent of \(-\frac{\pi}{4}\) is -1.

Step 3: Calculate \( \sin(-\frac{\pi}{4}) \)

Knowing that \( \theta = -\frac{\pi}{4} \), we now find \( \sin(-\frac{\pi}{4}) \). The sine of \(-\frac{\pi}{4}\) is \( -\frac{\sqrt{2}}{2} \) because the sine of a negative angle is negative and \( \sin\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} \).

Step 4: State the Result

Combine the insights from previous steps to state that \( \sin \left( \tan^{-1}(-1) \right) = -\frac{\sqrt{2}}{2} \).

Key Concepts

Tangent Function

The tangent function, denoted as \( \tan \), is a fundamental trigonometric function that plays a crucial role in right-angled triangles and circular motion. It is defined as the ratio of the opposite side to the adjacent side in a right-angled triangle. Understanding this ratio can make it easier to analyze angles and their relationships:

• Definition: For any angle \( \theta \), \( \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \).
• Inverse Tangent: The inverse tangent function, \( \tan^{-1}(x) \), helps us find an angle whose tangent is \( x \). This is commonly used to determine angles from known tangent values.
• Range of \( \tan^{-1} \): The principal value of \( \tan^{-1} \) is typically between \(-\frac{\pi}{2}\) and \(\frac{\pi}{2}\), ensuring we find a unique angle.

These properties make the tangent function incredibly useful for solving trigonometric equations and finding exact angle measures.

Sine Function

The sine function, \( \sin \), is another mainstay in trigonometry, measuring the vertical component of an angle in the unit circle approach. In a right-angled triangle, it is the ratio of the length of the opposite side to the hypotenuse:

• Definition: For an angle \( \theta \), \( \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \).
• Negative Angles: For negative angles, \( \sin(-\theta) = -\sin(\theta) \), which means that the sine function is odd.
• Special Angles: At \( \theta = \pm \frac{\pi}{4} \), the value is \( \pm \frac{\sqrt{2}}{2} \). This occurs because both sine and cosine are equal at these angles, reflecting symmetrical properties in circle quadrants.

Using the sine function in conjunction with inverse functions allows us to solve complex expressions by converting values into recognizable terms.

Angle Conversion

Angle conversion is the process of translating angles between different units or forms, like degrees and radians. It's a crucial skill for navigating problems in trigonometry. For our purposes, conversions often involve recognizing specific known angles and how they behave in different systems:

• Degrees and Radians: A full circle is \(360^\circ\) or \(2\pi\) radians. This means \(1^\circ = \frac{\pi}{180} \) radians.
• Quadrants: Angles can be positive or negative, depending on their quadrant. Negative angles are measured clockwise from the positive x-axis in the unit circle.
• Practical Conversion: In problems, it's common to see conversions, such as \( \tan^{-1}(-1) \), which converts to an angle \(-\frac{\pi}{4} \) with a known tangent value of -1.

Being adept at angle conversion helps in seamlessly moving between different trigonometric contexts and understanding the behavior of angles in various scenarios.