Problem 31 - Algebra and Trigonometry

Question

\(23-44=\) 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

\(-\frac{\pi}{6}\)

Step 1: Recognize the Expression

Identify the expression given: \( \sin^{-1}\left(\sin\left(-\frac{\pi}{6}\right)\right) \). This represents the inverse sine of the sine of \(-\frac{\pi}{6}\).

Step 2: Compute the Inner Sine

Compute \( \sin\left(-\frac{\pi}{6}\right) \). We know that \( \sin(-x) = -\sin(x) \), so \( \sin\left(-\frac{\pi}{6}\right) = -\sin\left(\frac{\pi}{6}\right) = -\frac{1}{2} \).

Step 3: Apply the Inverse Sine

Apply the inverse sine to the result from Step 2. We have \( \sin^{-1}(-\frac{1}{2}) \), which equals \(-\frac{\pi}{6}\) because \( -\frac{\pi}{6} \) is within the range \([-\frac{\pi}{2}, \frac{\pi}{2}]\) of the inverse sine function.

Step 4: Confirm the Result

Verify that the calculated \( -\frac{\pi}{6} \) is the correct principal value of the expression \( \sin^{-1}\left(\sin\left(-\frac{\pi}{6}\right)\right) \), as it lies in the principal range of the inverse sine function.

Key Concepts

Sine Function

The sine function is an essential trigonometric function in mathematics. It describes the relationship between an angle in a right triangle and the ratio of the length of the opposite side to the hypotenuse. Expressed as a function, it is written as \( \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \). This function is periodic, meaning it repeats its values in regular intervals, specifically every \( 2\pi \) radians or 360 degrees.
Key properties of the sine function:

• The domain of the sine function is all real numbers.
• The range is between -1 and 1.
• The sine function has specific symmetrical properties, such as \( \sin(-x) = -\sin(x) \).

These characteristics make the sine function a foundational tool for describing oscillatory motions, such as sound waves and light waves.

Principal Value

The principal value is the specific value of an inverse trigonometric function within a given range where the function is one-to-one. For the inverse sine, or arcsine, the principal range is typically \([-\frac{\pi}{2}, \frac{\pi}{2}]\).
Why is the principal value important?

• Ensures consistency in trigonometric calculations.
• Prevents multiple valid solutions by choosing a standard one.

In calculations, this means when you calculate \(\sin^{-1}(x)\), the result will always be within this interval. This restriction helps maintain a consistent and predictable outcome, which is especially useful in applications like physics and engineering.

Angle Measurement

Angles can be measured in degrees or radians, with radians being the standard unit in calculus and trigonometric functions. One radian equates to the angle formed when the arc length equals the radius of the circle. Consequently, \(2\pi\) radians equal a full circle, or 360 degrees.
Understanding angle measurement is crucial:

• It assists in correctly interpreting trigonometric functions.
• Allows conversion between different angular units, enhancing versatility in problem-solving.

For example, converting \(-\frac{\pi}{6}\) radians involves multiplying by \(\frac{180}{\pi}\), resulting in -30 degrees. Having a firm grasp on angle measurement aids in better comprehension of the contexts in which these functions are applied.

Inverse Sine Function

The inverse sine function, denoted as \(\sin^{-1}(x)\) or \( \text{arcsin}(x) \), is used to determine the angle whose sine value is \( x \). This function is restricted to inputs \([-1, 1]\), because those are the limits of the sine function's range.
Characteristics of the inverse sine function:

• The output, or the angle, is confined within \([-\frac{\pi}{2}, \frac{\pi}{2}]\).
• The function is the mirrored counterpart of the sine function over its principal range.
• Useful in solving equations with angles, especially when the angle needs to be extracted from a given sine ratio.

For example, solving \(\sin^{-1}(-\frac{1}{2})\) yields \(-\frac{\pi}{6}\), because \(-\frac{\pi}{6}\) is in the principal range, demonstrating how inverse sine helps reverse the effect of the sine function to retrieve the original angle.