Problem 26 - Algebra 2 and Trigonometry

Question

In \(15-26,\) find each exact value in radians, expressing each answer in terms of \(\pi\) \(y=\arcsin 0\)

Step-by-Step Solution

Answer

\(y = \arcsin 0 = 0\) radians.

Step 1: Understand the Definition

The arcsine function, denoted as \(\arcsin(x)\), is the inverse of the sine function. It returns the angle \(y\) in radians such that \(\sin(y) = x\) for \(y \in \left[-\frac{\pi}{2}, \frac{\pi}{2}\right]\). Here, we need to find \(y\) where \(\sin(y) = 0\).

Step 2: Identify Angle for Sine Value

To solve \(\sin(y) = 0\), we recall that the sine of an angle is zero at 0, \(\pi\), and multiples of \(\pi\). However, since the range of \(\arcsin\) is \([-\frac{\pi}{2}, \frac{\pi}{2}]\), we are restricted to finding \(y\) within this range.

Step 3: Determine the Exact Angle

Within the interval \([-\frac{\pi}{2}, \frac{\pi}{2}]\), the only angle where \(\sin(y) = 0\) is \(y = 0\). Thus, \(\arcsin(0) = 0\).

Step 4: Express the Result in Terms of Pi

Since \(y = 0\) is the solution, it is already expressed in terms of \(\pi\) as the radians equivalent of 0 is kept the same. Hence, the answer is \(0\) radians.

Key Concepts

Arcsin

The arcsine function, commonly written as \( \arcsin(x) \), is a type of inverse trigonometric function. Its purpose is to find an angle when the value of its sine is known. For instance, if you know \( \sin(y) = x \), then \( \arcsin(x) = y \). This is particularly useful because it allows us to reverse the process of finding sine, helping identify the angle that corresponds to a given sine value.

One key characteristic of the arcsine function is its range. For real numbers, \( \arcsin(x) \) produces an angle \( y \) that falls between \( -\frac{\pi}{2} \) and \( \frac{\pi}{2} \). This range is important because it ensures that the function remains a "least positive angle" solution, meaning the smallest angle that satisfies the equation. Whenever you deal with arcsine, remember the notation \( \arcsin(x) \) and how it helps in literally "finding the angle backward."

In practice, arcsine is often used to determine angles in triangles when side lengths are known, or in circular motion, to find the initial angle given some point’s position on a circle.

Sine Function

The sine function is a foundational concept in trigonometry. Typically written as \( \sin(y) \), it refers to a mathematical function that describes how an angle in a right triangle relates to the ratio of the side opposite that angle to the hypotenuse of the triangle.

Some fundamental properties of the sine function are:

• It is periodic, with a period of \( 2\pi \), meaning every \( 2\pi \) interval, the function values repeat.
• Its values range between \(-1\) and \(1\), as the opposite side cannot be longer than the hypotenuse.
• It is symmetric about the origin, exemplifying even-odd symmetry of trigonometric functions.
• Zero values occur at angles of \(0, \pi, 2\pi, \) etc.

The sine function helps in modeling oscillations, such as sound waves, light waves, and other forming periodic movements in science and engineering. Understanding how the sine function works become crucial, especially when progressing into more complex areas like physics or calculus.

Angle Measurement in Radians

Radians are a measure of angles used in many areas of mathematics, replacing the degree measurement commonly used in everyday life. Unlike degrees, which divide a circle into 360 parts, radians relate directly to the radius of a circle. One radian is the angle created when the arc length is equal to the radius of the circle.

This relation makes radians incredibly useful because they provide a more natural mathematical framework. For instance, a whole revolution around a circle equals \(2\pi\) radians, which reflects the circle’s circumference formula. Key angles often expressed in radians include:

• \(0 \) radians: The start of an angle, equivalent to \(0^\circ \).
• \( \frac{\pi}{2} \) radians: A right angle or \(90^\circ \).
• \(\pi\) radians: A straight angle or \(180^\circ \).
• \( \frac{3\pi}{2} \) radians: \(270^\circ \).
• \(2\pi\) radians: A full rotation or \(360^\circ \).

Using radians often simplifies calculations in calculus, primarily when working with trigonometric functions, because many formulas assume angles in radians for their natural derivation and integration properties.