Problem 114 - Precalculus

Question

For the following exercises, evaluate the expressions. $$ \sin ^{-1}\left(-\frac{1}{2}\right) $$

Step-by-Step Solution

Answer

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

Step 1: Understanding Inverse Sine

The inverse sine function, denoted as \( \sin^{-1}(x) \) or \( \arcsin(x) \), gives us the angle whose sine is \( x \). The range of \( \sin^{-1}(x) \) is from \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\) (or from \(-90^\circ\) to \(90^\circ\)).

Step 2: Identify the Known Angle

We need to find an angle \( \theta \) such that \( \sin(\theta) = -\frac{1}{2} \). The sine of an angle is \(-\frac{1}{2}\) at \( \theta = -\frac{\pi}{6} \) (or \(-30^\circ\)).

Step 3: Verify the Angle Within Range

The found angle \( \theta = -\frac{\pi}{6} \) is within the range \(-\frac{\pi}{2} \leq \theta \leq \frac{\pi}{2} \), which confirms it is a correct solution for the inverse sine function.

Key Concepts

Understanding Inverse Sine

Inverse trigonometric functions help us find angles when we know the value of a trigonometric ratio. The inverse sine function, represented as \(\sin^{-1}(x)\) or \(\arcsin(x)\), specifically provides the angle whose sine is \(x\). Unlike the regular sine function which gives a ratio, the inverse sine gives us an angle.

• It is important to remember that the range of the inverse sine function is limited. This means it only produces angles within a specific set of possible angles.
• The range for \(\sin^{-1}(x)\) is from \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\), which corresponds to angles from \(-90^\circ\) to \(90^\circ\).

This restriction helps to define a unique angle output for every valid sine value input. By understanding these principal values, you can effectively solve problems involving the sine inverse.

Angle Measurement in Trigonometry

When dealing with trigonometric functions, angles can be measured in degrees or radians. While the degree system is commonly used in everyday contexts, radians are preferred in mathematical calculations because they relate directly to the radius of a circle.

• The angle \(-30^\circ\), for instance, is equivalent to \(-\frac{\pi}{6}\) radians. This conversion is crucial when working with trigonometric equations.
• It is also useful to understand how these angles fall on the unit circle. For example, in radians, the full circle is \(2\pi\) which corresponds to \(360^\circ\).

When solving an inverse sine problem, you need to consider these conversions and representations of angles. A proper understanding of both these systems can help make sense of trigonometric functions and their inverses.

Trigonometric Range

The concept of range in trigonometry, especially in inverse functions, is crucial for determining the possible output values.

• For \(\sin^{-1}(x)\), the range is limited to \(\theta\) values where \(-\frac{\pi}{2} \le \theta \le \frac{\pi}{2}\), ensuring one definite angle result for each sine value \(x\).
• Understanding this range helps in identifying that every calculated angle using an inverse trigonometric function must fall within this specified range. Therefore, we can be sure that the angle \(-\frac{\pi}{6}\) (or \(-30^\circ\)) lies within allowable bounds for \(\sin^{-1}(x)\).

Knowing the range of the inverse sine aids in checking if your solution is valid, confirming that your answer not only matches the trigonometric equation but also fits the scope of possible results.