Problem 26 - Algebra and Trigonometry

Question

\(23-44=\) Find the exact value of the expression, if it is defined. \(\sin \left(\sin ^{-1} 5\right)\)

Step-by-Step Solution

Answer

The expression is undefined.

Step 1: Understand the Given Expression

The given expression is \( \sin(\sin^{-1}(5)) \). This might initially seem solvable as an inverse trigonometric function composition, but we must verify domain constraints for trigonometric functions.

Step 2: Examine Domain of \(\sin^{-1}(x)\)

The function \( \sin^{-1}(x) \), also known as arcsin, is defined only for \(-1 \leq x \leq 1\). It outputs values in the range of \([-\frac{\pi}{2}, \frac{\pi}{2}]\). Since 5 is not within the interval \([-1, 1]\), \( \sin^{-1}(5) \) is undefined.

Step 3: Conclusion: Determine the Expression's Value

Since \( \sin^{-1}(5) \) is undefined, the expression \( \sin(\sin^{-1}(5)) \) is also undefined because the composition involves an undefined component.

Key Concepts

Sine Function

In trigonometry, the sine function is a fundamental element that measures the y-coordinate of a point on the unit circle at a given angle. It is part of the basic trigonometric functions alongside cosine and tangent. The sine of an angle, often denoted as \( \sin(\theta) \), expresses how much the angle \( \theta \) deviates along the vertical axis from the origin of the circle.

Key properties of the sine function include:

• The sine function is periodic, meaning it repeats its values in regular intervals. Its period is \( 2\pi \).
• The range of the sine function lies between -1 and 1, as it represents a point on the unit circle, which never exceeds these boundaries.
• For angles between \(-\frac{\pi}{2}\) and \(\frac{\pi}{2}\), \( \sin(\theta) \) and \( \theta \) have a one-to-one relationship, making inverse calculations possible.

Understanding the sine function is crucial since it underlines many concepts in both geometry and calculus. It bridges angles and ratios, offering insights into periodic phenomena like sound waves and tide patterns.

Arcsin Domain

The inverse of the sine function, known as the arcsin or \( \sin^{-1}(x) \), reverses the sine's process by taking a ratio and returning the corresponding angle. This is essential in scenarios where one needs to determine the particular angle from a known sine value.

However, arcsin has a specific domain restriction:

• The domain of arcsin is \([-1, 1]\). This is because the sine function outputs only within this range.
• For inputs outside of this interval, such as 5 in our original exercise, arcsin is undefined as no angle can produce such a value on the unit circle.
• Correspondingly, the output range of arcsin is \([\-\frac{\pi}{2}, \frac{\pi}{2}]\), representing the angle \( \theta \) that satisfy the arcsin function.

These restrictions exist to ensure that the function remains invertible, meaning every input corresponds to only one output, which is critical for defining a function in mathematics.

Undefined Expressions

An expression may be deemed undefined in mathematics if it calls for a computation that has no meaning within its current operation context. In trigonometry, one typical example of an undefined expression involves inverse trigonometric functions with inputs outside their valid domain.

Here's why an expression becomes undefined:

• When you try to compute \( \sin^{-1}(5) \), the operation looks for an angle whose sine is 5. As this is beyond the sine range of \([-1,1]\), it is undefined.
• Consequently, the expression \( \sin(\sin^{-1}(5)) \) incorporates this undefined element, rendering the entire computation void.
• This underscores the importance of always checking constraints like domain and range before attempting to solve functions involving inverse calculations.

By understanding why an expression might be undefined, students can identify potential errors in trigonometric computations and ensure their work respects mathematical principles.