Problem 3 - Calculus Early Transcendentals: Pearson New International Edition

Question

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

Step-by-Step Solution

Answer

\(-\frac{\pi}{3}\)

Step 1: Understand the Problem

We need to find the angle \( \theta \) such that \( \sin(\theta) = -\frac{\sqrt{3}}{2} \) and \( \theta \) is within the range of \( \sin^{-1} \), which is \([-\frac{\pi}{2}, \frac{\pi}{2}]\).

Step 2: Recall Sine Values for Special Angles

Recall that \( \sin\left(-\frac{\pi}{3}\right) = -\frac{\sqrt{3}}{2} \) because \( \sin \left( \frac{\pi}{3} \right) = \frac{\sqrt{3}}{2} \). This falls within the valid range for \( \sin^{-1} \) and satisfies the equation.

Step 3: Check the Range

Ensure that the solution \( -\frac{\pi}{3} \) lies within \([-\frac{\pi}{2}, \frac{\pi}{2}]\). Since \( -\frac{\pi}{3} \) is between \(-\frac{\pi}{2}\) and \( \frac{\pi}{2} \), it is a valid answer.

Step 4: Verify Solution

Double-check that substituting \(-\frac{\pi}{3}\) back into the sine function gives the correct value: \( \sin\left(-\frac{\pi}{3}\right) = -\frac{\sqrt{3}}{2} \). Thus, our answer is verified.

Key Concepts

sine function

The sine function is a crucial component in trigonometry, often represented as \( \sin(\theta) \), where \( \theta \) is the angle. It calculates the ratio of the length of the opposite side to the hypotenuse in a right triangle. The values of the sine function vary from \(-1\) to \(1\). This means the sine is always a positive or negative fractional number within this range. The inverse sine function, written as \( \sin^{-1}(x) \), or "arc sine", aims to find the angle \( \theta \) when the sine value is known. It has a range of \([ -\frac{\pi}{2}, \frac{\pi}{2} ]\), meaning it only provides angle results within these specific limits. This is necessary because sine is periodic and would otherwise provide multiple answers. Having a good understanding of the sine function helps in accurately working with angles and lengths in triangles, which then supports solving problems involving inverse trigonometric functions.

special angles

Special angles in trigonometry include angles like \(0\), \(\frac{\pi}{6}\), \(\frac{\pi}{4}\), \(\frac{\pi}{3}\), and \(\frac{\pi}{2}\), and their negatives. These angles are "special" because their trigonometric functions have known exact values.For the sine function, specifically:

• \(\sin(0) = 0\)
• \(\sin(\frac{\pi}{6}) = \frac{1}{2}\)
• \(\sin(\frac{\pi}{4}) = \frac{\sqrt{2}}{2}\)
• \(\sin(\frac{\pi}{3}) = \frac{\sqrt{3}}{2}\)
• \(\sin(\frac{\pi}{2}) = 1\)

These known values are what make it possible to easily solve problems like finding the inverse sine without a calculator. Knowing these angles and their sine values helps quickly recognize that \(\sin\left(-\frac{\pi}{3}\right) = -\frac{\sqrt{3}}{2}\). This is because the negative angle reflects the same height in the opposite direction along the unit circle.

trigonometric identities

Trigonometric identities are equations that hold true for all angles. They are incredibly useful for simplifying complex trigonometric equations and confirming solutions. Some fundamental identities relevant to the sine function include:

• Pythagorean Identity: \(\sin^2(\theta) + \cos^2(\theta) = 1\)
• Negative Angle Identity: \(\sin(-\theta) = -\sin(\theta)\)

In the solved exercise, the negative angle identity is particularly relevant. It explains why \(\sin\left(-\frac{\pi}{3}\right)\) can be swiftly calculated as \(-\frac{\sqrt{3}}{2}\). This identity shows that the sine of a negative angle is simply the negative of the sine of the corresponding positive angle. Using these identities enhances your ability to tackle various problems by understanding the relationships between the trigonometric functions and their operations.