Problem 21 - A Graphical Approach to Precalculus with Limits

Question

Find the exact value of each real number \(y .\) Do not use a calculator. $$y=\sin ^{-1} \frac{\sqrt{2}}{2}$$

Step-by-Step Solution

Answer

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

Step 1: Understand the Problem

We need to find the angle \( y \) whose sine value is \( \frac{\sqrt{2}}{2} \). This means \( y \) is equal to \( \sin^{-1} \left( \frac{\sqrt{2}}{2} \right) \).

Step 2: Recall the Definition of Inverse Sine

The expression \( \sin^{-1}(x) \) provides the angle \( y \) such that \( -\frac{\pi}{2} \leq y \leq \frac{\pi}{2} \) and \( \sin(y) = x \).

Step 3: Identify Angles with Given Sine Value

The output \( \frac{\sqrt{2}}{2} \) is a common sine value for the angles \( \frac{\pi}{4} \) and its equivalents. However, within the restricted range of \( \sin^{-1} \), we focus on the angle \( \frac{\pi}{4} \), where \( \sin \left( \frac{\pi}{4} \right) = \frac{\sqrt{2}}{2} \).

Step 4: Verify the Solution

Verify that \( \sin \left( \frac{\pi}{4} \right) \) indeed equals \( \frac{\sqrt{2}}{2} \), confirming that \( y = \frac{\pi}{4} \) is the correct solution.

Key Concepts

Sine Function

The sine function is one of the fundamental trigonometric functions in mathematics. It arises commonly in situations involving angles and triangulation. The sine function, often denoted as \( \sin \theta \), represents the ratio of the length of the opposite side to the hypotenuse in a right triangle. Its value varies as the angle \( \theta \) changes.

• The sine function is periodic with a period of \( 2\pi \), meaning it repeats every \( 2\pi \) radians or 360 degrees.
• Its range is between -1 and 1, inclusive.
• It is an odd function, which implies that \( \sin(-\theta) = -\sin(\theta) \).

When considering inverse trigonometric situations, such as finding an angle whose sine is \( \frac{\sqrt{2}}{2} \), the sine function helps identify possible angle solutions within controlled limits, as seen in the inverse sine or arcsine function.

Exact Values of Trigonometric Functions

Exact values of trigonometric functions are often encountered in the realm of angles that are integral parts of the unit circle. These exact values can simplify problems that require precise solutions, like those in the inverse trigonometric functions.

• Common angles include \( \frac{\pi}{6} \), \( \frac{\pi}{4} \), and \( \frac{\pi}{3} \). For these, the sine values are precisely known: \( \sin \frac{\pi}{6} = \frac{1}{2} \), \( \sin \frac{\pi}{4} = \frac{\sqrt{2}}{2} \), \( \sin \frac{\pi}{3} = \frac{\sqrt{3}}{2} \).
• Exact values allow mathematicians to work without approximations, maintaining precision in calculations.

These exact values simplify calculations in many mathematical problems by providing a reliable reference for common angle measurements, eliminating the need for a calculator.

Angle Measurement

Angle measurement is a crucial part of trigonometry and geometry. It dictates how angles are quantified and is essential for determining values of trigonometric functions.

• There are different units for measuring angles: degrees and radians.
• The choice of measurement can influence the calculations involved in trigonometric equations.

In calculations involving inverse trigonometric functions, angle measurements become particularly important as they define the range of possible solutions. Knowing whether to calculate in degrees or radians is crucial for obtaining the correct answer, as each system has distinct properties and uses.

Radians

Radians serve as an alternative to degrees for measuring angles. This unit is not based on divisions of a circle into parts like degrees, but rather on the radius of the circle itself.

• One radian is defined to be the angle created when the arc length equals the radius length.
• A complete circle is \( 2\pi \) radians.
• They provide a natural way to express angles in mathematical calculations, particularly when dealing with periodic functions and calculus.

In trigonometry, using radians often simplifies various expressions and calculations. For instance, it allows for seamless integration and differentiation involving trigonometric functions. In the context of inverse functions, recognizing radians as the unit of choice for angles ensures accurate and practical solutions, as demonstrated in problems like finding \( y = \sin^{-1}(\frac{\sqrt{2}}{2}) \), where the solution is conveniently given as \( \frac{\pi}{4} \).