Problem 24 - A Graphical Approach to Precalculus with Limits

Question

Find the exact value of each real number \(y .\) Do not use a calculator. $$y=\arcsin \left(-\frac{\sqrt{2}}{2}\right)$$

Step-by-Step Solution

Answer

The exact value of \( y \) is \(-\frac{\pi}{4}\).

Step 1: Identify the Range of arcsin Function

The function \( \arcsin(x) \) returns values in the range \([ -\frac{\pi}{2}, \frac{\pi}{2} ]\). This means that for \(y = \arcsin(x)\), \(y\) is determined between \(-\frac{\pi}{2}\) and \(\frac{\pi}{2}\).

Step 2: Determine Sine Value and Reference Angle

Given that \( \sin(y) = -\frac{\sqrt{2}}{2} \), recall the angles for which the sine value is \( \frac{\sqrt{2}}{2} \) are \( \frac{\pi}{4} \) and \(-\frac{\pi}{4} \). Since the value is negative here, focus on \(-\frac{\pi}{4} \) because it's in the range of \([-\frac{\pi}{2}, \frac{\pi}{2}]\).

Step 3: Assign Exact Value of y Based on Sign

Since \(-\frac{\sqrt{2}}{2}\) is negative and the range of \( \arcsin(x) \) is from \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\), the exact value for \(y\) is \(-\frac{\pi}{4}\). Thus, \(y = \arcsin \left(-\frac{\sqrt{2}}{2}\right) = -\frac{\pi}{4}\).

Key Concepts

Trigonometric Functions

Trigonometric functions are essential for understanding relationships between angles and sides in right triangles. These functions allow you to calculate unknown values in trigonometry and cover sine, cosine, tangent, and more. In a right triangle, the sine of an angle is the ratio of the length of the opposite side to the hypotenuse. Similarly, cosine and tangent are other ratios involving the sides of a triangle:

• Cosine is the ratio of the adjacent side to the hypotenuse.
• Tangent is the ratio of the opposite side to the adjacent side.

In the unit circle, a circle with radius 1, the sine value of an angle corresponds to the y-coordinate of a point on the circle, while the cosine corresponds to the x-coordinate. This understanding is crucial because it allows you to solve problems like finding angles or side lengths when restricted to a specific domain.

Inverse Trigonometric Functions

Inverse trigonometric functions are the counterparts of the original trigonometric functions. They are used to find angles when given a trigonometric value. The terms used are arcsin, arccos, and arctan, for sine, cosine, and tangent respectively. For example, while the sine function provides the ratio, the arcsin function determines the angle that corresponds to a specific ratio.

• Arcsin returns an angle whose sine is the given value.
• Arccos finds the angle whose cosine is that value.
• Arctan calculates the angle whose tangent matches the input value.

These functions are restricted to particular ranges to ensure they remain functions. For example, the range of the arcsin function is \( [-\frac{\pi}{2}, \frac{\pi}{2}] \), which is necessary for providing a unique and predictable output for each input in its domain.

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. The unit circle allows you to visualize trigonometric functions and understand their values. As you rotate around the circle, the coordinates of points on the circle tell you the values of cosine and sine for various angles. This makes it easier to work with trigonometric identities and solve problems.

• The coordinates \((x, y)\) on the unit circle represent \( (\cos(\theta), \sin(\theta)) \).
• The angle \( \theta \) is typically measured from the positive x-axis.

The unit circle also helps to understand the ranges of inverse trigonometric functions. For example, arcsin is confined to angles between \( -\frac{\pi}{2} \) and \( \frac{\pi}{2} \), since these correspond to the y-values on the top and bottom halves of the unit circle. This tool is invaluable for solving trigonometric equations and visualizing their solutions.