Problem 15 - Algebra 2 and Trigonometry

Question

In \(15-26,\) find each exact value in radians, expressing each answer in terms of \(\pi\) \(y=\arcsin 1\)

Step-by-Step Solution

Answer

The exact value is \( \frac{\pi}{2} \).

Step 1: Understanding the Problem

We need to find the exact value of the inverse sine function, also known as arcsine, for the input value of 1. The function arcsine gives the angle whose sine is the given number.

Step 2: Recall the Definition of Arcsine

The arcsine function, \( ext{arcsin}(x)\), returns an angle \(y\) within the interval \([-\frac{\pi}{2}, \frac{\pi}{2}]\) such that \( ext{sin}(y) = x\).

Step 3: Solve for the Angle

We are looking for an angle \(y\) such that \( ext{sin}(y) = 1\). We know that \(\text{sin}\left(\frac{\pi}{2}\right) = 1\).

Step 4: Determine the Exact Value

Since the sine of \(\frac{\pi}{2}\) is 1, and \(\frac{\pi}{2}\) falls within the range of arcsine, the exact value of \(y = \text{arcsin}(1)\) is \(\frac{\pi}{2}\).

Key Concepts

Understanding Arcsine

Imagine you're trying to figure out which angle, when its sine is calculated, equals a given number. This is exactly what the arcsine function does! Arcsine, often written as \( \text{arcsin}(x) \), is the inverse of the sine function. So if you know the sine of an angle, arcsine will help you find the angle itself. This concept is particularly handy when you want to "reverse" a sine operation.

• **Range:** The arcsine function only returns values between \(-\frac{\pi}{2}\) and \(\frac{\pi}{2}\). This means the angles you get as a result of the arcsine are always within these bounds.
• **Usage:** When you see something like \(y = \text{arcsin}(x)\), it's asking "what angle \(y\) has a sine of \(x\), while still being between \(-\frac{\pi}{2}\) and \(\frac{\pi}{2}\)?"
• **Arcsin for 1:** For example, if we want \(\text{arcsin}(1)\), we're looking for the angle whose sine is 1, and which falls in our valid range.

The Role of Radians

Radians are a way of expressing angles that are very convenient for mathematical calculations. Instead of using degrees, which cut circles into 360 parts, radians deal with the radius of the circle. One full circle around uses the circle’s radius to measure out the distance, leading to roughly 6.28 radians, or more precisely, 2\(\pi\) radians.

• **Why Radians?** Using radians often makes calculations simpler, especially in calculus. Functions involving sine and cosine behave more predictably and intuitively. For example, the sine of \( \frac{\pi}{2} \) is simply one.
• **Converting Degrees to Radians:** Remember, \(180^\circ \) is \(\pi\) in radians. This relationship helps in converting back and forth between these two systems.
• **Special Angles:** Commonly used angles, like \(\frac{\pi}{2} \), \(\pi\), and \(\frac{3\pi}{2} \), are staples. They’re much like the landmarks in the realm of degrees (like 90° or 180°).

Finding Exact Values

When we talk about "exact values" in trigonometry, we mean the specific, precise values of trigonometric functions without any approximation. These are often expressed using radicals (\(\sqrt{}\)), integers, or values of \(\pi\).

• **No Decimals:** Exact values avoid decimals, which might approximate a number. Instead, they give a clean, accurate representation.
• **Common Trigonometric Values:** For the arcsine function specifically, knowing that \(\text{sin}(\frac{\pi}{2}) = 1\) helps. It means \(\text{arcsin}(1) = \frac{\pi}{2}\), all done without touching a calculator!
• **Importance in Calculations:** The certainty of exact values is essential in higher mathematics, where precision is key.