Problem 1 - Contemporary Precalculus

Question

Find the exact functional value without using a calculator: $$\sin ^{-1} 1$$

Step-by-Step Solution

Answer

Answer: The exact functional value of the given expression, $$\sin^{-1}(1)$$, is $$\frac{\pi}{2}$$.

Step 1: Recall the properties of the inverse sine function

The sine function, $$\sin (x)$$, has a corresponding inverse function, $$\sin ^{-1}(x)$$, which is only defined in the principal range, $$[-\frac{\pi}{2}, \frac{\pi}{2}]$$. We are looking for the angle $$x$$, such that its sine value is 1.

Step 2: Recall the definition of the inverse sine function

The definition of $$\sin^{-1}(x)$$ is that for any $$x$$, $$\sin(\sin^{-1}(x)) = x$$. Using this definition, we need to find an angle in the principal range that has a sine value equal to 1.

Step 3: Identify the angle in the principal range that has a sine value equal to 1

We know that the maximum value of the sine function is 1 and it occurs at angle $$\frac{\pi}{2}$$. Since $$\frac{\pi}{2}$$ is in the principal range we were looking for, we can conclude that $$\sin^{-1}(1)=\frac{\pi}{2}$$.

Step 4: Write the final answer

Thus, the exact functional value of the given expression, $$\sin^{-1}(1)$$, is equal to $$\frac{\pi}{2}$$.

Key Concepts

Sine Function

The sine function is one of the fundamental trigonometric functions, often denoted as \( \sin(x) \). It represents the y-coordinate of a point on the unit circle at a given angle \( x \). The function is periodic with a period of \( 2\pi \), meaning that \( \sin(x) = \sin(x + 2k\pi) \) for any integer \( k \).

Some key characteristics of the sine function include:

• It takes on values between -1 and 1 for all real numbers \( x \).
• It reaches its maximum value of 1 at \( \frac{\pi}{2} + 2k\pi \), where \( k \) is an integer.
• It reaches its minimum value of -1 at \( \frac{3\pi}{2} + 2k\pi \).

These properties make it a vital tool for modeling oscillatory and wave-like behaviors in various fields, from physics to engineering.

Principal Range

When working with inverse trigonometric functions like the inverse sine function \( \sin^{-1}(x) \), it's important to understand the concept of the principal range. The principal range is the interval within which the inverse function is defined and gives us a single, unique angle for any sine value.

For the sine function, the principal range of \( \sin^{-1}(x) \) is \([-\frac{\pi}{2}, \frac{\pi}{2}]\). This range was chosen because it includes the important feature of monotonicity (the function is always increasing), and it covers every possible value of the sine function from -1 to 1.
Understanding the principal range helps ensure that inverse function solutions like \( \sin^{-1}(1) \) are unique and consistent, without multiple possibilities or ambiguities.

Exact Functional Value

Determining the exact functional value of an expression like \( \sin^{-1}(1) \) involves finding the angle whose sine is equal to the given number—in this case, 1. Using our knowledge of the sine function and its properties, we look within the principal range of \([-\frac{\pi}{2}, \frac{\pi}{2}]\) to identify the angle.

For \( \sin^{-1}(1) \), we locate the particular angle within this range which produces the value 1 when evaluated in the sine function. This angle is \( \frac{\pi}{2} \), where the sine function achieves its peak value. Therefore, the exact functional value is \( \frac{\pi}{2} \).
These exact values are incredibly useful in calculus and precalculus courses, as they form the backbone of many fundamental identities and simplify complex equations without reaching for a calculator.