Problem 69 - Precalculus with Limits

Question

Advanced Applications In Exercises 69 and 70 , solve the system of equations for \(u\) and \(v .\) While solving for these variables, consider the transcendental functions as constants. (Systems of this type appear in a course in differential equations.) $$ \left\\{\begin{array}{l}{u \sin x+v \cos x=0} \\ {u \cos x-v \sin x=\sec x}\end{array}\right. $$

Step-by-Step Solution

Answer

After following the outlined steps, the solution for \(u\) is \(u = - \sqrt{\frac{1}{1 + \tan^2x}}\) and for \(v\) is \(v = \tan{x}u\).

Step 1: Square and Add

First, square both of the given equations and then add them together. This gives us \(u^2(\sin^2x + \cos^2x) + v^2(\cos^2x + \sin^2x) = \sec^2x\)

Step 2: Simplify

Next, simplify the equation using the identity \(\sin^2x + \cos^2x = 1\), resulting in the equation \(u^2 + v^2 = \sec^2x\) or \(u^2 + v^2 = 1/\cos^2x\).

Step 3: Substitute Solution

Rearranging the second equation, we get \(v = \frac{(u\cos x-\sec x)}{\sin x}\). Substitute this into the simplified equation from Step 2 and derive the solution for \(u\) and \(v\) respectively.

Key Concepts

Trigonometric Identities

Trigonometric identities are fundamental truths about trigonometric functions that hold true for all values of the variables involved. In this exercise, one crucial identity in play is the Pythagorean identity: \( \sin^2x + \cos^2x = 1 \).

This identity asserts that the square of the sine of an angle plus the square of the cosine of the same angle always equals one. It is at the heart of simplifying equations that involve sine and cosine. Using this identity, you can tackle many trigonometric problems, making complex expressions more manageable.

Here, this identity helps to simplify the system of equations by transforming complex combinations of trig functions into a simpler numeric form.

• They allow transformations like \( u^2(\sin^2x + \cos^2x) = u^2 \)
• They often lead to solutions that involve standard functions, like solving for \( u \) and \( v \) in this exercise.

Knowing and applying identities effectively aids in unraveling system of equations in mathematics and physics.

Transcendental Functions

Transcendental functions, such as sine and cosine, appear in equations and systems due to their non-algebraic nature. These functions are essential in modeling periodic phenomena. In many cases, transcendental functions are treated as constants when isolating other variables.

In this exercise, the transcendental functions are in the equations involving trigonometric identities. It is important to remember that while they behave predictably, their solutions may not always be straightforward.

• Transcendental functions include logarithmic, exponential, sinusoidal, and hyperbolic functions.
• They differ from polynomial equations because their solutions can't be expressed using finite algebraic operations.
• They frequently appear in calculus, physics, engineering, and other scientific calculations.

Handling these functions effectively requires familiarity with their properties and the ability to manipulate them in conjunction with algebraic operations.

Differential Equations

Differential equations involve functions and their derivatives and are integral to modeling how quantities change. Although the given exercise doesn't directly require solving differential equations, it presents a system resembling those found when dealing with such equations.

Systems like these might, for example, appear when determining a system's stability or behavior over time. Understanding how to simplify and solve equations with such systems helps interpret results from differential equations.

Some characteristics of differential equations include:

• Use of rates of change to describe real-world systems over time.
• Appearance in physics, engineering, biology, and economics.
• Solutions can predict how systems evolve, providing insights into stability or oscillatory behavior.

Mastery of solving systems of equations, and using transcendental functions and identities, will enhance comprehension and application of differential equations in more complex scenarios.