Problem 7 - Precalculus

Question

Find the exact value. \(\arcsin \left(\frac{\sqrt{2}}{2}\right)\)

Step-by-Step Solution

Answer

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

Step 1: Understand the Function

The function we have is \(\arcsin\left(\frac{\sqrt{2}}{2}\right)\). The \(\arcsin\) function gives us the angle whose sine is \(\frac{\sqrt{2}}{2}\).

Step 2: Recall the Sine Values for Special Angles

Recall that the sine of \(45^\circ\) or \(\frac{\pi}{4}\, \text{radians}\) is \(\frac{\sqrt{2}}{2}\). This means the angle with a sine of \(\frac{\sqrt{2}}{2}\) is \(\frac{\pi}{4}\) radians.

Step 3: Check the Range of Arcsine Function

The range of the \(\arcsin\) function is \([-\frac{\pi}{2}, \frac{\pi}{2}]\). Since \(\frac{\pi}{4}\) is within this range, it is the correct value for \(\arcsin\left(\frac{\sqrt{2}}{2}\right)\).

Key Concepts

Arcsine

The arcsine function, often denoted as \( \arcsin(x) \), is one of the inverse trigonometric functions. It helps to ascertain the angle whose sine is \( x \). For example, \( \arcsin \left(\frac{\sqrt{2}}{2}\right) \) is an exercise to find the angle whose sine equals \( \frac{\sqrt{2}}{2} \). This function is particularly useful when working with angles and trigonometric identities, providing essential links between different elements of trigonometry.

In mathematical terms, when you see \( \arcsin(x) \), it asks, 'What angle has a sine of \( x \)?' Furthermore, the range of the arcsine function is restricted to \([ -\frac{\pi}{2}, \frac{\pi}{2} ]\) to ensure that it reliably produces only one value: the principal value, making it more predictable to work with. This specific range mirrors the increasing segment of the sine function, which bridges smoothly into our journey through trigonometry.

Special Angles

Special angles are specific angles with known sine, cosine, and tangent values, making them highly useful for solving trigonometric problems efficiently. They often include angles like \(30^\circ\), \(45^\circ\), and \(60^\circ\), which correspond to \(\frac{\pi}{6}\), \(\frac{\pi}{4}\), and \(\frac{\pi}{3}\) radians respectively.

In the context of our problem, knowing that the sine of \(45^\circ\) (or \(\frac{\pi}{4}\) radians) is \(\frac{\sqrt{2}}{2}\) immediately helps us determine that \(\arcsin \left(\frac{\sqrt{2}}{2}\right)\) equates to \(\frac{\pi}{4}\). Utilizing these angles simplifies problem-solving because their trigonometric ratios are standardly taught and memorized, thus acting as a vital tool in learning and applying trigonometry.

Sine Function

The sine function, usually represented as \( \sin(\theta) \), is a fundamental trigonometric function that expresses the y-coordinate of a point on the unit circle corresponding to a given angle \( \theta \). This function oscillates between -1 and 1, with the domain of all real numbers, and it is periodic with a period of \( 2\pi \).

In the example of our problem, the function \( \sin \left(\frac{\pi}{4}\right) \) yields \( \frac{\sqrt{2}}{2} \), showing the real-world applicability of the sine function. Such concrete instances demonstrate how angles and their corresponding coordinates are interlinked, setting the basis for a wide array of applications in mathematics and beyond, from wave behavior to circle-related geometry.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all angles in the identities' domains. These identities, like the Pythagorean identity \( \sin^2(\theta) + \cos^2(\theta) = 1 \), are incredibly useful for simplifying complex trigonometric expressions.

In our exploration of the arcsine of \( \frac{\sqrt{2}}{2} \), understanding the identity of \( \sin(\theta) = \cos \left(\frac{\pi}{2} - \theta \right) \) becomes vital in expressing and deducing relationships between angles. Mastery of such identities opens the door to more sophisticated analysis in trigonometry and applied mathematics, enabling a deeper understanding of relationships between angles, sides, and phenomena modeled by trigonometric functions.