Problem 110 - A Graphical Approach to Precalculus with Limits

Question

Write each expression as an algebraic expression in \(u, u>0\). $$\sec \left(\cos ^{-1} \frac{u}{\sqrt{u^{2}+5}}\right)$$

Step-by-Step Solution

Answer

The expression is \( \frac{\sqrt{u^2+5}}{u} \).

Step 1: Understand the Problem

We need to express \( \sec \left( \cos^{-1} \frac{u}{\sqrt{u^2+5}} \right) \) in terms of \( u \). The term \( \cos^{-1} \left( \frac{u}{\sqrt{u^2+5}} \right) \) represents an angle, and the expression requires us to find the secant of this angle.

Step 2: Use Trigonometric Identity

Recall that if \( \theta = \cos^{-1} x \), then \( \cos\theta = x \). Therefore, for our problem, we have \( \cos \theta = \frac{u}{\sqrt{u^2+5}} \), where \( \theta = \cos^{-1} \left( \frac{u}{\sqrt{u^2+5}} \right) \). We need \( \sec \theta \) which is \( \frac{1}{\cos \theta} \).

Step 3: Calculate \( \sec \theta \)

The secant function is the reciprocal of cosine. So, \( \sec \theta = \frac{1}{\cos \theta} \). Substitute \( \cos \theta = \frac{u}{\sqrt{u^2+5}} \) into this equation to get \( \sec \theta = \frac{1}{\frac{u}{\sqrt{u^2+5}}} = \frac{\sqrt{u^2+5}}{u} \).

Step 4: Final Expression

The expression for \( \sec \left( \cos^{-1} \frac{u}{\sqrt{u^2+5}} \right) \) can now be written as \( \frac{\sqrt{u^2+5}}{u} \) in terms of \( u \), with the condition \( u > 0 \).

Key Concepts

Trigonometric Identities

Trigonometric identities are essential tools in trigonometry and are often used to simplify expressions and solve equations. These identities are equations that hold true for all values of the variable where both sides of the equation are defined.

• The Pythagorean identities, such as \( \sin^2 \theta + \cos^2 \theta = 1 \), are fundamental in understanding how sine and cosine relate to each other.
• Reciprocal identities, like \( \sec \theta = \frac{1}{\cos \theta} \), are also crucial as they express one trigonometric function in terms of another.
• Understanding these identities allows us to transform complex expressions into simpler forms, like replacing trigonometric expressions with their algebraic equivalents.

In our exercise, recognizing that the secant function is the reciprocal of cosine helped transform the original expression into an algebraic one. This step is a classic application of trigonometric identities to simplify trigonometric expressions.

Inverse Trigonometric Functions

Inverse trigonometric functions are used to find the angle that corresponds to a given trigonometric value. They are the inverse operations of the basic trigonometric functions like sine, cosine, and tangent. When you see a notation like \( \cos^{-1}(x) \), it indicates the angle whose cosine is \( x \).

• The range of \( \cos^{-1}(x) \) is between 0 and \( \pi \) radians since the function returns the principal value of the angle.
• This concept is crucial when working with angles, especially when the trigonometric ratios are given, and one must determine the angle.
• In the original problem, \( \cos^{-1} \left( \frac{u}{\sqrt{u^2+5}} \right) \) gives us an angle \( \theta \), which the secant function then uses.

Mastering inverse trigonometric functions involves understanding their domains and ranges, which restrict the values they can take. This particular exercise delicately links the inverse cosine to the secant function to express the angle algebraically through the variable \( u \).

Secant Function

The secant function, written as \( \sec \), is one of the less commonly used trigonometric functions but remains important. It is the reciprocal of the cosine function, making it indispensable in specific problems. This function is defined as \( \sec \theta = \frac{1}{\cos \theta} \) wherever \( \cos \theta eq 0 \).

• The secant function is essential when dealing with expressions that require reciprocal identities.
• In math problems, converting secant into other trigonometric forms can simplify difficult trigonometric expressions.
• In our exercise, using \( \sec \theta = \frac{1}{\cos \theta} \) was key to transitioning from a trigonometric form to a usable algebraic form, resulting in \( \frac{\sqrt{u^2+5}}{u} \).

Getting comfortable with the secant function involves understanding how it interacts with other trigonometric functions and identities. It often pops up in contexts where a reciprocal identity can clear up complex relationships within an equation.