Problem 69 - Precalculus: A Concise Course

Question

Write an algebraic expression that is equivalent to the expression. (Hint: Sketch a right triangle, as demonstrated in Example \(7 .)\) $$ \cos (\arcsin 2 x) $$

Step-by-Step Solution

Answer

The algebraic expression equivalent to \(\cos (\arcsin 2x)\) is \(\sqrt{1 - (2x)^2}\)

Step 1: Apply Pythagorean Identity

The value of \(\sin \theta\) can be given as \(2x\). Consequently, apply the Pythagorean identity \(\cos^2 \theta = 1 - \sin^2 \theta\) with \(\theta = \arcsin 2x\). Thus, \(\cos^2 (\arcsin 2x) = 1 - (2x)^2\).

Step 2: Taking Square Roots

Since cosine function can have positive and negative values depending on the interval of \(\theta\), we obtain two possible solutions: \(\cos (\arcsin 2x) = \sqrt{1 - (2x)^2}\) and \(\cos (\arcsin 2x) = -\sqrt{1 - (2x)^2}\).

Step 3: Determining the Appropriate Interval

To select the correct equation, consideration of the interval for arcsine is critical. The range for \(\arcsin x\) is \(- \frac{\pi}{2} \leq x \leq \frac{\pi}{2}\). Within this range, the cosine function is non-negative. Hence, we discard the negative root.

Key Concepts

Understanding Algebraic Expression

An algebraic expression is a mathematical phrase that can contain numbers, operators (like +, -, *, /), and one or more variables (like x, y, z) representing values that can change. Simply put, it's like a recipe that tells you how to mix these different mathematical ingredients to create something new.

An algebraic expression does not contain an equality sign; that's what sets it apart from an equation. For example, in the problem \( \cos(\arcsin 2x) \), we have such an expression. We're looking to rewrite it in a way that's simpler or more useful for certain types of analysis. To understand this expression better, it's helpful to know about the functions involved and the relationships between them, which brings us into the realm of trigonometry.

Diving Into Trigonometric Functions

Trigonometric functions are a mathematician's tools for analyzing the properties of triangles, particularly right-angled triangles. These functions — sine (sin), cosine (cos), tangent (tan), and their reciprocals: cosecant (csc), secant (sec), and cotangent (cot) — relate the angles of a triangle to the lengths of its sides.

These functions don't just work with angles measured in degrees; they're also linked with radians, a different measure of angles. Trigonometric functions have a wide range of applications, from solving simple geometric problems to more complex calculations in physics, engineering, and beyond.

Exploring Arcsin and Cosine Relationship

The relationship between the arcsin function (the inverse of sine) and cosine is grounded in the trigonometric identities that these functions follow. Inverse trigonometric functions, like arcsin, return the angle whose trigonometric function equals the given number.

For the instance of \( \cos(\arcsin 2x) \), we're essentially looking for a cosine of an angle which, when you take the sine, gives you 2x. Since we know the sine and cosine functions are related via the Pythagorean identity \( \sin^2\theta + \cos^2\theta = 1 \) for any angle \(\theta\), we can use this relationship to reframe the algebraic expression in terms of only one trigonometric function.

Inverse Trigonometric Functions Unveiled

Inverse trigonometric functions are the bread and butter when it comes to finding angles when we know the sides. Take arcsin as an example: This function will tell you the angle whose sine is a specific value. The key to understanding these functions is knowing their ranges and what they imply.

Inverse functions are always accompanied by a principal range. This means that they give only one value (angle) for a given input. For the solution process in our example, we consider that \(\arcsin\) has a range from \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\), and the associated cosine values within this interval are always non-negative. This knowledge is crucial to choosing the right solution among the possibilities.