Problem 23 - Algebra and Trigonometry

Question

\(23-44=\) Find the exact value of the expression, if it is defined. \(\sin \left(\sin ^{-1} \frac{1}{4}\right)\)

Step-by-Step Solution

Answer

The exact value is \(\frac{1}{4}\).

Step 1: Understand the Problem

The problem asks us to evaluate \( \sin \left( \sin^{-1} \frac{1}{4} \right) \). Here, \( \sin^{-1} \) is the inverse sine function, also known as arcsin, which gives us an angle whose sine is the given value.

Step 2: Apply Inverse Trigonometric Properties

The property of inverse trigonometric functions states that \( \sin(\sin^{-1}(x)) = x \) for values of \( x \) in the range \([-1, 1]\). Since \( \frac{1}{4} \) is between \(-1\) and \(1\), it is within the domain of the inverse sine function.

Step 3: Simplify the Expression

Using the property from Step 2, we directly simplify \( \sin(\sin^{-1}(\frac{1}{4})) \) to \( \frac{1}{4} \). This is because the inverse sine, \( \sin^{-1}(\frac{1}{4}) \), outputs an angle whose sine is \( \frac{1}{4} \), and taking the sine of that angle returns \( \frac{1}{4} \) itself.

Key Concepts

Inverse Sine Function

The inverse sine function, often denoted as \( \sin^{-1} \) or arcsin, is a special function in trigonometry. Its purpose is to find an angle when we know the sine of that angle.

Let's break it down simplistically:

• If you have a sine value, arcsin will tell you the angle whose sine is that value.
• It only works for sine values between -1 and 1, as sine cannot exceed these limits.
• The resulting angle from \( \sin^{-1} \) usually falls within the range of \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\), or -90 to 90 degrees, to ensure it is uniquely defined.

The inverse sine is quite handy for solving trigonometric equations where you need to find an angle. Whenever you see an arc such as arcsin(x), it asks "what angle has a sine of x?"

It's crucial to remember that while \( \sin^{-1}(x) \) refers to the angle, taking the sine of this angle will return to your original x, given x is within the domain.

Arcsin

Arcsin, represented as the inverse sine function \( \sin^{-1} \), describes the same concept but in a slightly different notation. Its usage is primarily interchangeable with inverse sine.

Here are the key points about the arcsin function:

• Arcsin provides the measure of angles when the sine value is known.
• This function is typically discussed in the contexts of right triangles and unit circles.
• Arcsin angles are real numbers for all sine values between \(-1\) and \(1\).

Consider its role in relation to the unit circle: any sine value corresponds directly to a point on this circle. For instance, \( \sin^{-1}(\frac{1}{4}) \) locates an angle in the first quadrant with that sine value.

Remembering arcsin's properties helps in recognizing that \( \sin(\sin^{-1}(x)) \) effectively recovers the original sine value x.

Trigonometric Properties

Trigonometric properties are foundational rules that apply to functions like sine and its inverse, arcsin. These properties help us solve problems and understand the relationships between the functions.

Some basic properties to consider:

• The sine of an inverse sine function \( \sin(\sin^{-1}(x)) = x \) for \( x \) in the range \([-1, 1]\). This confirms that you're taking a sine value back to what it represents: an angle.
• In contrast, \( \sin^{-1}(\sin(x)) = x \) holds for \(x\) values specifically within the fundamental range \(-\frac{\pi}{2} \leq x \leq \frac{\pi}{2}\).
• Inverse trigonometric functions like arcsin have specific ranges to maintain function properties like one-to-one mapping and ensuring each output has a unique pre-image.

These properties not only depict how trigonometric functions transition from angles to values and back, but they also underscore the importance of correctly understanding their domains and ranges.

Mastering these properties enables you to decode complex expressions and solve trigonometric equations efficiently.