Problem 26 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 23-30: Write the expression as an algebraic expression in \(x\) for \(x>0\). $$ \cot \left(\sin ^{-1} \frac{\sqrt{x^{2}-9}}{x}\right) $$

Step-by-Step Solution

Answer

The expression is \( \frac{3}{\sqrt{x^2-9}} \).

Step 1: Understanding the Problem

We need to rewrite the given trigonometric expression \( \cot \left(\sin^{-1} \frac{\sqrt{x^2-9}}{x}\right) \) as an algebraic expression in terms of \( x \) where \( x>0 \). This involves using trigonometric identities and relationships.

Step 2: Identifying Components

First, recognize that \( \sin^{-1} \frac{\sqrt{x^2-9}}{x} \) represents an angle \( \theta \), such that \( \sin \theta = \frac{\sqrt{x^2-9}}{x} \). Our goal is to find \( \cot \theta \), which is \( \frac{\cos \theta}{\sin \theta} \).

Step 3: Applying Pythagorean Identity

Since \( \sin \theta = \frac{\sqrt{x^2-9}}{x} \), we use the Pythagorean identity \( \sin^2 \theta + \cos^2 \theta = 1 \) to find \( \cos \theta \). Substitute \( \sin \theta \) into the identity: \( \left(\frac{\sqrt{x^2-9}}{x}\right)^2 + \cos^2 \theta = 1 \).

Step 4: Simplifying the Expression

Calculate \( \left(\frac{\sqrt{x^2-9}}{x}\right)^2 = \frac{x^2-9}{x^2} \). Substitute this into the identity: \( \frac{x^2-9}{x^2} + \cos^2 \theta = 1 \). Solve for \( \cos^2 \theta \): \( \cos^2 \theta = 1 - \frac{x^2-9}{x^2} = \frac{9}{x^2} \).

Step 5: Finding \( \cos \theta \)

Since \( \cos^2 \theta = \frac{9}{x^2} \), we have \( \cos \theta = \frac{3}{x} \) for \( x > 0 \), ensuring \( \cos \theta \) is positive as \( \theta \) is in the first quadrant.

Step 6: Calculating \( \cot \theta \)

Now, \( \cot \theta = \frac{\cos \theta}{\sin \theta} = \frac{\frac{3}{x}}{\frac{\sqrt{x^2-9}}{x}} = \frac{3}{\sqrt{x^2-9}} \).

Step 7: Expressing Final Solution

Therefore, the algebraic expression for \( \cot \left(\sin^{-1} \frac{\sqrt{x^2-9}}{x}\right) \) in terms of \( x \) is \( \frac{3}{\sqrt{x^2-9}} \).

Key Concepts

Trigonometric Identities

Trigonometric identities are crucial for simplifying expressions and solving equations involving trigonometric functions. These identities include relationships between angles and side lengths in a right triangle. Common identities include:

• Pythagorean Identities: These involve relationships such as \( \sin^2 \theta + \cos^2 \theta = 1 \), which is used to relate sine and cosine of an angle.
• Reciprocal Identities: For example, \( \cot \theta = \frac{1}{\tan \theta} \) or \( \csc \theta = \frac{1}{\sin \theta} \).
• Angle Sum and Difference Identities: These help find the sine, cosine, or tangent of a sum or difference of two angles.

When dealing with more complex trigonometric expressions such as \( \cot \left(\sin^{-1} \frac{\sqrt{x^2-9}}{x}\right) \), it's essential to know how to apply these identities correctly. By breaking down expressions into simpler parts, such as identifying an angle using inverse trigonometric functions, these identities can help convert a trigonometric expression into an algebraic one.

Inverse Trigonometric Functions

Inverse trigonometric functions are used to find the angle that corresponds to a given trigonometric ratio. For instance, \( \sin^{-1}(y) \) gives the angle whose sine is \( y \). These functions help in transforming trigonometric expressions into algebraic ones.

• Range and Domain: Inverse trigonometric functions often have specific ranges and domains. For \( \sin^{-1}(y) \), the range is \([-\frac{\pi}{2}, \frac{\pi}{2}]\) and the domain is \([-1, 1]\).
• Application: In our exercise, we start with \( \theta = \sin^{-1} \frac{\sqrt{x^2-9}}{x} \), meaning \( \sin \theta = \frac{\sqrt{x^2-9}}{x} \). This step is key as it transitions the expression into something a trigonometric identity can work with.

Understanding these functions is essential as they often serve as a bridge in converting trigonometric equations into algebraic form, offering a straightforward path to simplification.

Pythagorean Identity

The Pythagorean identity is one of the most vital trigonometric identities used to relate the sine and cosine of an angle. It states that \( \sin^2 \theta + \cos^2 \theta = 1 \). This identity allows us to solve for one function if we know the other.In our exercise, by knowing \( \sin \theta = \frac{\sqrt{x^2-9}}{x} \), we can use the Pythagorean identity to find \( \cos \theta \):- Substitute \( \sin \theta \) into the identity: \[ \left(\frac{\sqrt{x^2-9}}{x}\right)^2 + \cos^2 \theta = 1 \]- Simplify the expression: \[ \frac{x^2-9}{x^2} + \cos^2 \theta = 1 \]- Solve for \( \cos^2 \theta \): \[ \cos^2 \theta = \frac{9}{x^2} \]Thus, we can find \( \cos \theta \) as \( \frac{3}{x} \) for \( x > 0 \), where \( \cos \theta \) must be positive since \( \theta \) is in the first quadrant. Finally, this identity simplifies our task of calculating \( \cot \theta = \frac{\cos \theta}{\sin \theta} \), leading to the final algebraic expression.