Problem 36 - Precalculus with Calculus Previews

Question

Write the given expression as an algebraic expression in \(x\). $$ \sec (\arccos x) $$

Step-by-Step Solution

Answer

The expression is \( \frac{1}{x} \).

Step 1: Understanding the Inverse Trigonometric Function

The expression given is \( \sec(\arccos(x)) \). Here, \( \arccos(x) \) represents the angle \( \theta \) such that \( \cos(\theta) = x \).

Step 2: Relating Trigonometric Functions

We know that \( \sec(\theta) = \frac{1}{\cos(\theta)} \). Hence, \( \sec(\arccos(x)) = \frac{1}{\cos(\theta)} = \frac{1}{x} \). Therefore, initially it seems like \( \sec(\arccos(x)) = \frac{1}{x} \).

Step 3: Using Pythagorean Identity

Recall the Pythagorean identity \( \sin^2(\theta) + \cos^2(\theta) = 1 \). Since \( \cos(\theta) = x \), it follows that \( \sin^2(\theta) = 1 - x^2 \) and consequently \( \sin(\theta) = \sqrt{1 - x^2} \).

Step 4: Finding Secant from Cosine

Since \( \sec(\theta) = \frac{1}{\cos(\theta)} \), we substitute the value of \( \cos(\theta) \) which is \( x \). Hence, \( \sec(\theta) = \frac{1}{x} \).

Step 5: Conclusion

Therefore, the expression \( \sec(\arccos(x)) \) simplifies directly to \( \frac{1}{x} \) when expressed algebraically in \( x \).

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions are essential tools in mathematics for solving angles when you know the value of their trigonometric functions. Take the function \( \arccos(x) \) as an example. \( \arccos(x) \) returns the angle \( \theta \) such that \( \cos(\theta) = x \). This means if you have a cosine value, you can find out the actual angle with \( \arccos \). These functions reverse the trigonometric functions:

• \( \arcsin(x) \) returns an angle such that \( \sin(\theta) = x \).
• \( \arctan(x) \) returns an angle such that \( \tan(\theta) = x \).

Understanding how these work is crucial for transforming expressions like \( \sec(\arccos(x)) \) into something much simpler. By knowing that \( \arccos(x) = \theta \), where \( \cos(\theta) = x \), you're able to rewrite the expression using trigonometric identities.

Pythagorean Identity

The Pythagorean identity is a fundamental concept in trigonometry. It states that for any angle \( \theta \), \( \sin^2(\theta) + \cos^2(\theta) = 1 \). This identity is incredibly useful when you have one of the trigonometric values but need to find another.In the context of \( \sec(\arccos(x)) \), it's important to know that if \( \cos(\theta) = x \), you can find the sine value using:

• \( \sin^2(\theta) = 1 - x^2 \)
• This implies \( \sin(\theta) = \sqrt{1 - x^2} \), considering the angle is within the standard range for the inverse functions.

Therefore, Pythagorean identity helps in finding other trigonometric values necessary for simplifying expressions.

Algebraic Expression

Converting trigonometric expressions into simple algebraic expressions can make solving them easier. An algebraic expression involves operations like addition, subtraction, multiplication, and division combined with variables like \( x \). To simplify \( \sec(\arccos(x)) \), understanding how each function relates is critical:

• The secant of an angle \( \theta \) is given by \( \sec(\theta) = \frac{1}{\cos(\theta)} \).
• If you have \( \arccos(x) = \theta \), you directly know \( \cos(\theta) = x \). This transforms the secant function to \( \sec(\theta) = \frac{1}{x} \).

By simplifying trigonometric expressions to algebraic forms, such as replacing \( \sec(\arccos(x)) \) with \( \frac{1}{x} \), you make calculations more manageable, paving the way for easier problem-solving. Recognizing this transformation allows you to work with equations efficiently.