Problem 7 - Calculus Early Transcendentals: Pearson New International Edition

Question

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

Step-by-Step Solution

Answer

The exact value is \(-\frac{\pi}{6}\).

Step 1: Understanding the Problem

The problem asks us to find the exact value of the inverse sine function, also known as the arcsine function, evaluated at \(-\frac{1}{2}\). This means we are looking for an angle \(\theta\) such that \( \sin(\theta) = -\frac{1}{2} \) and \(\theta\) lies within the arcsine function's range, which is \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\).

Step 2: Using Known Values of Sine

We start by recalling the sine values for common angles. We know from the unit circle that \( \sin(\frac{\pi}{6}) = \frac{1}{2} \). To find an angle that results in \(-\frac{1}{2}\), we can look at angles in the fourth quadrant where the sine value is negative.

Step 3: Finding the Exact Angle

Since \( \sin(\theta) = -\frac{1}{2} \) and the sine of \( \frac{\pi}{6} \) is \( \frac{1}{2} \), the angle \( \theta \) that corresponds to \( \sin(\theta) = -\frac{1}{2} \) is in the fourth quadrant. It is \( \theta = -\frac{\pi}{6} \) because \( \sin(-\frac{\pi}{6}) = -\sin(\frac{\pi}{6}) = -\frac{1}{2} \).

Step 4: Verify the Angle is within Range

The arcsine function, \( \arcsin(x) \), gives angles within the interval \([-\frac{\pi}{2}, \frac{\pi}{2}]\). The angle \(-\frac{\pi}{6}\) belongs to this interval, which confirms that it is a valid solution for \(\arcsin(-\frac{1}{2})\).

Key Concepts

Arcsine Function

The arcsine function, denoted as \( \arcsin(x) \), is the inverse of the sine function. It is used to determine an angle whose sine is a given number. This function is particularly useful when you have the sine of an angle and need to find the angle itself. The range of the arcsine function is restricted to ensure that it produces unique outputs. It spans from \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\).
This means that any angle provided as a result of an arcsine operation will lie within this interval.

• The primary purpose of the arcsine function is to return the angle for a given sine value.
• It is crucial in situations where the exact angle is needed, particularly in trigonometry and calculus.
• The range limitation is necessary to maintain the function's single-valued nature.

In doing calculations like \( \arcsin\left(-\frac{1}{2}\right) \), it is important to consider the negative sign. The angle found \(\theta = -\frac{\pi}{6}\) is well within the permissible range for arcsine.

Unit Circle

The unit circle is a fundamental concept in trigonometry. It is a circle with a radius of 1, centered at the origin of a coordinate plane. Each point on the unit circle represents an angle and its corresponding sine and cosine values.
The position on the circle is defined by the angle made with the positive x-axis.

• For the unit circle, the value of the sine function corresponds to the y-coordinate of a point.
• The cosine function corresponds to the x-coordinate.

The unit circle is exceedingly useful in solving trigonometric equations as it provides a geometric representation of the functions. To solve \( \arcsin(-\frac{1}{2}) \), we look for an angle where the y-coordinate (sine value) is \(-\frac{1}{2}\). This occurs in the fourth quadrant, leading to \(\theta = -\frac{\pi}{6}\). Understanding the unit circle's layout helps to visualize and solve such problems more intuitively.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values of the variables involved. They are crucial for simplifying expressions and solving trigonometric equations.

• Basic identities involve relations between sine, cosine, and tangent functions.
• They include the Pythagorean identity: \( \sin^2(\theta) + \cos^2(\theta) = 1 \).

In our context, knowing that \( \sin(-\theta) = -\sin(\theta) \) helps us find the sine of negative angles. This identity was applied when finding that \( \arcsin(-\frac{1}{2}) \) corresponds to the angle \(-\frac{\pi}{6}\).Additionally, the knowledge of sine values for key angles like \(\frac{\pi}{6}\), \(\frac{\pi}{4}\), and \(\frac{\pi}{3}\) allows quick identification of angles corresponding to specific sine values. Trigonometric identities provide essential tools for handling complex trigonometric expressions and calculations.