Problem 45 - Algebra and Trigonometry

Question

Find the exact value of each expression, if possible. Do not use a calculator. $$ \sin \left(\sin ^{-1} \pi\right) $$

Step-by-Step Solution

Answer

The given expression is not defined due to the input of the inverse sine function being outside its domain.

Step 1: Understand the Inverse Sine Function

The inverse sine function, also known as arcsine, undoes the operation of the sine function. In other words, considering a number \( x \), the sine operation can change this \( x \) to a specific output value, let's call it \( y \). The inverse sine operation is able to turn this output value \( y \) back into the original number \( x \).

Step 2: Apply the Inverse Sine Function

In this exercise, we have \( \sin (\sin ^{-1} \pi) \), with \( \pi \) being the input to the inverse sine function. The output of \( \sin^{-1} \pi \) is not defined because the range of the sine function lies in the interval \(-1 \leq y \leq 1\), and \( \pi \) is not in this interval. Therefore, the exercise as given does not have a solution.

Key Concepts

Understanding the Sine Function

The sine function is a fundamental concept in trigonometry and works to relate angles of a right triangle to the lengths of its sides. For a given angle in a right triangle, the sine of that angle is the ratio of the length of the opposite side to the hypotenuse. This relation can also be expressed in a more general form for angles beyond the first quadrant by using the unit circle.

• The sine function is periodic with a cycle of \( 2\pi \).
• It is used extensively in various fields such as physics, engineering, and even music.

Understanding the sine function's behavior and properties helps form the foundation for learning its inverse, known as arcsine.

Exploring the Arcsine

The arcsine function, denoted as \( \sin^{-1}(x) \), provides the angle whose sine is a given number. Arcsine essentially reverses the sine function. It's important to note that while the sine can take any angle as input, arcsine is more restricted.

• The output of arcsine is always an angle, specifically within the range \(-\frac{\pi}{2}\) to \( \frac{\pi}{2} \).
• This restriction allows arcsine to be a function, meaning it will always return a single, real-valued angle for any input within its domain.

The arcsine function is critical for solving equations where angles need to be deduced from the values of sine.

Range of the Sine Function

The range of the sine function is crucial for understanding when the arcsine is defined. The sine function takes any real number as input, but its range is constrained between -1 and 1.

• This means the sine function's output will always be within the interval \([-1, 1]\).
• When dealing with inverse functions like arcsine, understanding this range is necessary to grasp what inputs arcsine can handle.

In exercises involving inverse trigonometric functions, checking if the input falls within this valid range is an essential first step.

Understanding Undefined Expressions in Trigonometry

In trigonometry, undefined expressions occur when operations are attempted with inputs that lie outside of valid ranges. For the arcsine function, this happens when we try to compute it for any value not between -1 and 1.

• If you attempt to calculate \( \sin^{-1}(\pi) \), it results in an undefined expression since \( \pi \) is not within the acceptable range for arcsine.
• As with many mathematical concepts, understanding why certain expressions are undefined helps avoid errors and confusion.

When solving problems, always ensure that the expressions are defined within their respective function's domain.