Problem 19 - Algebra and Trigonometry

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

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

Step 1: Understand the Inverse Sine Function

The inverse sine function, also known as arcsine and denoted as \( \sin^{-1} \), returns the angle whose sine is a given number. Its range is \(-\frac{\pi}{2} \leq \theta \leq \frac{\pi}{2}\).

Step 2: Evaluate the Inner Sine

Calculate \( \sin \left(-\frac{\pi}{6}\right) \). Since sine is an odd function, \( \sin(-x) = -\sin(x) \). Therefore, \( \sin \left(-\frac{\pi}{6}\right) = -\sin \left(\frac{\pi}{6}\right) = -\frac{1}{2}\).

Step 3: Apply the Inverse Sine Function

Now, we need to find \( \sin^{-1} \left(-\frac{1}{2}\right) \), which is the angle in the range \(-\frac{\pi}{2} \leq \theta \leq \frac{\pi}{2}\) whose sine value is \(-\frac{1}{2}\). The angle within this range is \(-\frac{\pi}{6}\).

Step 4: Conclusion

Thus, \( \sin^{-1}(\sin(-\frac{\pi}{6})) = -\frac{\pi}{6} \). This result is defined since \(-\frac{\pi}{6}\) is within the range of the inverse sine function.

Key Concepts

Trigonometric Identities

Trigonometric identities are fundamental relationships between the angles and sides of a triangle. They help simplify trigonometric expressions and solve complex problems. One key identity is the property of odd and even functions. The sine function, for example, is an odd function. This means that \( \sin(-x) = -\sin(x) \). This property helps us understand transformations of the sine function, such as flips over the x-axis.
Another essential identity is the Pythagorean identity, which for sine and cosine is expressed as \( \sin^2(x) + \cos^2(x) = 1 \). This identity is useful for converting sine values into cosine values, and vice versa. Recognizing these identities enables seamless transitions between different trigonometric forms.

• Sine and cosine are periodic functions, repeating every \(2\pi\) radians.
• Being familiar with these identities aids in solving equations and evaluating expressions.

Understanding these identities lays the groundwork for tackling more complex trigonometric problems.

Sine Function

The sine function is pivotal in trigonometry and arises frequently in geometry, physics, and engineering. It is defined for a real number \( x \), and returns a value between -1 and 1, representing the y-coordinate of a point on the unit circle corresponding to an angle \( x \).
The sine of an angle in a right triangle represents the ratio of the length of the side opposite the angle to the length of the hypotenuse. When graphed, the sine function produces a smooth, wave-like curve known as a sinusoidal wave.

• The sine wave starts at zero, reaches a maximum value at \( \frac{\pi}{2} \), returns to zero at \( \pi \), reaches a minimum at \( \frac{3\pi}{2} \), and returns to zero at \( 2\pi \).
• This periodic nature depicts the regular cycling of the sine values.

Understanding the sine function is crucial for working with waves and oscillations in various scientific fields.

Range of Arcsine

The inverse sine function, known as arcsine and denoted as \( \sin^{-1} \), provides the angle whose sine is a given number between -1 and 1. The range of arcsine is the interval \(-\frac{\pi}{2} \leq y \leq \frac{\pi}{2}\). This range signifies the possible angles returned by the arcsine function, which are angles in the first and fourth quadrants of the unit circle.
When you apply \( \sin^{-1} \) to a sine value, such as \( -\frac{1}{2} \), the result will be an angle within this range. It ensures that the arcsine function is single-valued and continuous.

• Angles obtained from arcsine are always measured in radians unless otherwise specified.
• This range is crucial for defining the principal value of the inverse sine function.

By keeping to its specified range, the arcsine function effectively communicates its input angle's sine value within the confines of a meaningful mathematic interval.