Problem 26 - Precalculus: Functions and Graphs

Question

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 simplifies to \( \frac{3}{\sqrt{x^2-9}} \).

Step 1: Recognize the Inverse Function

We have the expression \( \cot \left(\sin^{-1} \frac{\sqrt{x^{2}-9}}{x}\right) \). The function \( \sin^{-1} \) indicates the inverse of sine, so if \( \theta = \sin^{-1} \frac{\sqrt{x^{2}-9}}{x} \), then \( \sin(\theta) = \frac{\sqrt{x^{2}-9}}{x} \).

Step 2: Set the Trigonometric Identity

Recall the Pythagorean identity: \( \sin^2(\theta) + \cos^2(\theta) = 1 \). Knowing that \( \sin(\theta) = \frac{\sqrt{x^{2}-9}}{x} \), we can write: \( \sin^2(\theta) = \frac{x^2 - 9}{x^2} \).

Step 3: Solve for Cosine

Using \( \sin^2(\theta) = \frac{x^2 - 9}{x^2} \), substitute it into the Pythagorean identity: \( \frac{x^2 - 9}{x^2} + \cos^2(\theta) = 1 \). Solve for \( \cos^2(\theta) \): \( \cos^2(\theta) = \frac{9}{x^2} \), thus \( \cos(\theta) = \frac{3}{x} \) since \( x > 0 \).

Step 4: Determine the Cotangent

Recall that \( \cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)} \). Substituting the expressions for \( \cos(\theta) = \frac{3}{x} \) and \( \sin(\theta) = \frac{\sqrt{x^2 - 9}}{x} \), we get: \[ \cot(\theta) = \frac{\frac{3}{x}}{\frac{\sqrt{x^2 - 9}}{x}} = \frac{3}{\sqrt{x^2 - 9}}. \]

Step 5: Simplify the Expression

The simplified expression for \( \cot \left( \sin^{-1} \frac{\sqrt{x^2-9}}{x} \right) \) is \( \frac{3}{\sqrt{x^2-9}} \).

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions are the reverse operations of the basic trigonometric functions like sine, cosine, and tangent. They help us find the angle when the value of the trigonometric function is given. For instance, if we know that \( \sin(\theta) = \frac{\sqrt{x^{2}-9}}{x}\), we can find the angle \(\theta\) using the inverse sine function, denoted as \(\sin^{-1}\). This tells us the angle whose sine is \frac{\sqrt{x^{2}-9}}{x}\. When you see \(\sin^{-1}\), it's requesting the measure of the angle that satisfies the given sine value.

• Inverse sine is also called arcsin.
• For \(\sin^{-1}(y)\), ensure that \(-1 \leq y \leq 1\).
• Helps express complex trigonometric calculations as algebraic expressions.

By understanding the role of inverse trigonometric functions, converting their results into algebraic expressions like in this exercise becomes more manageable.

Algebraic Expression

An algebraic expression involves numbers, variables, and operation symbols. When solving trigonometric equations, converting trigonometric expressions into algebraic ones can simplify the solving process, as is the case in this exercise. We initially have \( \cot\left(\sin^{-1} \frac{\sqrt{x^{2}-9}}{x}\right)\). To express this entirely in terms of x, we use our knowledge of trigonometric identities and basic algebra.
Steps taken include:

• Identifying the trigonometric components, such as \( \sin^{-1} \), of the angle.
• Using trigonometric identities to find \( \sin(\theta)\) and \( \cos(\theta)\).
• Substituting back to express quantities like \( \cot(\theta)\) in terms of x.

This results in simpler, often non-trigonometric expressions, which can be easier to analyze and manipulate. Transforming complex trigonometric expressions into algebraic forms is a powerful tool in calculus and algebra, simplifying analysis and problem-solving.

Pythagorean Identity

The Pythagorean identity is one of the basic integral identities used in trigonometry, relating the sine and cosine of an angle. It's presented as \(\sin^2(\theta) + \cos^2(\theta) = 1\). This is fundamental to trigonometric calculations and transformations. It expresses the relationship between the sides of a right triangle and is derived from the Pythagorean Theorem.
This identity is especially useful when:

• Converting between trigonometric expressions.
• Finding unknown trigonometric values.
• Simplifying complex trigonometric proofs and expressions.

In the solution, we used the Pythagorean identity to solve for \( \cos(\theta)\) given \( \sin(\theta) = \frac{\sqrt{x^{2}-9}}{x}\). By substituting into the identity, it allowed us to find \( \cos^2(\theta)\), and ultimately \( \cos(\theta)\). This step is pivotal in simplifying the expression to its final form in algebraic terms.

Cotangent

Cotangent, written as \( \cot(\theta)\), is a trigonometric function that is the reciprocal of the tangent function. Moreover, it's defined as the ratio of the adjacent side to the opposite side in a right triangle, or equivalently, \( \frac{\cos(\theta)}{\sin(\theta)}\). In contexts like this exercise, understanding cotangent helps express angles concerning other trigonometric functions in algebraic forms.
Some key points about cotangent are:

• \( \cot(\theta) = \frac{1}{\tan(\theta)}\).
• It complements tangent in such a way that \(\cot^2(\theta) + 1 = \csc^2(\theta)\), a form of Pythagorean identity.
• Often used to simplify expressions with tangent.

In the original exercise, after determining \( \cos(\theta)\) and \( \sin(\theta)\), we used the cotangent formula to transform the expression into its final simplified form, which is \( \frac{3}{\sqrt{x^2-9}}\), a neat and convenient algebraic expression.