Problem 31 - Precalculus Mathematics for Calculus

Question

Find the exact value of the expression, if it is defined. $$\sin ^{-1}\left(\sin \left(-\frac{\pi}{6}\right)\right)$$

Step-by-Step Solution

Answer

The exact value is \(-\frac{\pi}{6}\).

Step 1: Evaluate the Inner Sine Function

The first step is to evaluate the function inside the inverse sine. We start with \( \sin \left(-\frac{\pi}{6}\right) \). From trigonometry, \( \sin(-x) = -\sin(x) \). So, we have: \( \sin \left(-\frac{\pi}{6}\right) = -\sin \left(\frac{\pi}{6}\right) = -\frac{1}{2} \).

Step 2: Apply the Inverse Sine Function

Now that we have \( \sin^{-1}(-\frac{1}{2}) \), we need to determine the angle in the range \([-\frac{\pi}{2}, \frac{\pi}{2}]\) for which the sine is \(-\frac{1}{2}\). The angle \(-\frac{\pi}{6}\) has a sine of \(-\frac{1}{2}\), and it lies within this range.

Step 3: Determine Exact Value

Since the angle \(-\frac{\pi}{6}\) is already in the principal range of the inverse sine function, we conclude that \( \sin^{-1}(\sin(-\frac{\pi}{6})) = -\frac{\pi}{6} \).

Key Concepts

Sine Function

The sine function is one of the fundamental trigonometric functions, which associates angles to specific ratios. More specifically, it relates the angle in a right-angled triangle to the opposite side divided by the hypotenuse. When exploring in terms of the unit circle, the sine function can be defined as the y-coordinate of the point where the terminal side of the angle intersects the unit circle. This circle has a radius of 1 centered at the origin of the coordinate plane. The sine function is periodic with a fundamental period of \(2\pi\), meaning that it repeats every \(2\pi\) radians, which is equivalent to 360 degrees. If you graph the sine function, you'd observe a smooth, wave-like curve that crosses the x-axis at integer multiples of \(\pi\).

• The sine function is typically denoted as \( \sin(x) \).
• The range of the sine function is \([-1, 1]\), which corresponds to its maximum and minimum values.
• The sine function is odd, meaning \( \sin(-x) = -\sin(x) \), which is crucial in solving exercises such as this one.

Understanding these characteristics aids in assessing the behavior of the sine function over its various domains.

Angle Evaluation

Angle evaluation involves determining the corresponding lengths or ratios like sine for a given angle, typically measured in radians or degrees. In trigonometry, it often requires using identities or known values to simplify the given expressions.In this exercise, evaluating the inner sine function involves understanding how angles in different quadrants affect the sine value. The angle \(-\frac{\pi}{6}\), for instance, is equivalent to \(-30\) degrees. By recognizing that the sine function is odd, we understand that evaluating \(\sin(-x)\) results in \(-\sin(x)\). Therefore, \(\sin(-\frac{\pi}{6}) = -\sin(\frac{\pi}{6})\).

• The reference angle \(\frac{\pi}{6}\) (or 30 degrees) has a known sine value of \(\frac{1}{2}\).
• By applying the property \(\sin(-x) = -\sin(x)\), we easily find that \(\sin(-\frac{\pi}{6}) = -\frac{1}{2}\).

Such evaluations are helpful in not only solving problems but also in validating solutions by understanding the behavior of trigonometric functions over the principal and symmetric domains.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all angles. These identities encompass relationships between the basic trigonometric functions like sine, cosine, and tangent.One crucial identity used in our exercise is \(\sin(-x) = -\sin(x)\), reflecting the odd nature of the sine function. Trigonometric identities can also include relationships between functions like the Pythagorean identity \( \sin^2(x) + \cos^2(x) = 1 \) or angle-sum identities like \( \sin(x + y) = \sin x\cos y + \cos x\sin y \).

• These identities simplify complex trigonometric functions into more manageable expressions.
• They are essential for manipulating and solving various mathematical and real-world problems.

Understanding and applying these identities can lead to a deeper comprehension of how trigonometric functions interact and can be transformed, which is particularly valuable in simplifying and evaluating expressions in exercises like the one we solved.