Problem 3 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 1-10, find the exact value without using a calculator. $$ \sin ^{-1}\left(-\frac{\sqrt{3}}{2}\right) $$

Step-by-Step Solution

Answer

The exact value is \(-\frac{\pi}{3}\).

Step 1: Understanding the Problem

The problem asks us to find the exact value of the inverse sine, also known as arcsine, of \(-\frac{\sqrt{3}}{2}\). The arcsine function, denoted as \(\sin^{-1}(x)\), returns an angle whose sine is \(x\). The range of the sine inverse function is \([-\frac{\pi}{2}, \frac{\pi}{2}]\).

Step 2: Recalling the Sine Values

We need to recall the common sine values within the standard range of the arcsine function. The sine of specific angles is commonly memorized: \(\sin\left(-\frac{\pi}{3}\right) = -\frac{\sqrt{3}}{2}\). These values will help us identify the angle that corresponds to \(-\frac{\sqrt{3}}{2}\).

Step 3: Matching the Sine Value

Since the sine of \(-\frac{\pi}{3}\) is \(-\frac{\sqrt{3}}{2}\), then \(\sin^{-1}\left(-\frac{\sqrt{3}}{2}\right) = -\frac{\pi}{3}\). Ensure the angle is within the interval of \([-\frac{\pi}{2}, \frac{\pi}{2}]\), which it is, satisfying the condition for the \text{arcsin function range.

Key Concepts

Arcsine

The concept of arcsine, denoted as \( \sin^{-1}(x) \), relates closely to the sine function, but instead of taking an angle and giving you its sine value, it does the reverse. The arcsine function accepts a sine value and returns the corresponding angle. This angle will typically fall within the primary range of the arcsine function, which is \([-\frac{\pi}{2}, \frac{\pi}{2}]\). This ensures the angles produced are within the first or fourth quadrants of the unit circle.

To understand this better, visualize the graph of the sine function, which is periodic. Since repeating the same pattern can give multiple angles with the same sine value, for arcsine to be a function with a unique output for each input, we restrict it to this principal range. For instance, when we determine that \(\sin^{-1}\left(-\frac{\sqrt{3}}{2}\right) = -\frac{\pi}{3}\), we are using the reference angle \(\frac{\pi}{3}\) within the correct range to find the exact angle that the arcsine will return.

Trigonometric Identities

Trigonometric identities are powerful tools in calculating various trigonometric values, solving equations, and simplifying expressions. These identities are equations that hold true for any angle. Fundamental identities include the Pythagorean identity, angles sum identities, and double-angle formulas, among others.

Understanding these identities can immensely help in problems where one needs to work with various trigonometric functions, and they often serve as steps in deducing inverse trigonometric functions like arcsine. For example, a basic identity like \( \sin^2(\theta) + \cos^2(\theta) = 1\) can help relate sine and cosine values, which might be necessary in verifying certain sine values presented in different contexts.

• Using identities can help confirm whether an angle value given satisfies specific trigonometric conditions.
• The identity \( \sin(-x) = -\sin(x) \) can directly tell us the relationship of negative angles with their positive counterparts.

Exact Trigonometric Values

Exact trigonometric values of certain angles are essential to know because they frequently appear in exercises and exams. These include well-known angles like \(0, \frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{3}, \frac{\pi}{2}\), and their negatives.

For solving inverse trigonometric problems, having these values in your toolkit allows for quick mental calculations rather than working them out each time from scratch. For example, memorizing that \( \sin\left(-\frac{\pi}{3}\right) = -\frac{\sqrt{3}}{2} \) speeds up solving exercises like ours.

• These values are derived from what we know about the unit circle and the symmetrical properties of sine and cosine functions.
• Practicing the derivation and usage of these values helps in seeing patterns and relationships between different trigonometric angles and their ratios.

Learning these exact values provides confidence in tackling more complex trigonometric problems and in the classification of solutions.