Problem 109 - Calculus Early Transcendentals

Question

Prove that the equations $$x=a \cos t+b \sin t, \quad y=c \cos t+d \sin t,$$ where \(a, b, c,\) and \(d\) are real numbers, describe a circle of radius \(R\) provided \(a^{2}+c^{2}=b^{2}+d^{2}=R^{2}\) and \(a b+c d=0.\)

Step-by-Step Solution

Answer

Question: Prove that the parametric equations \(x = a \cos t + b \sin t\) and \(y = c \cos t + d \sin t\) represent a circle with radius R, given the conditions \(a^{2}+c^{2}=b^{2}+d^{2}=R^{2}\) and \(a b+c d=0\). Answer: To prove that the given parametric equations describe a circle with radius R, we followed these steps: 1. We started with the given parametric equations: \(x = a \cos t + b \sin t\) and \(y = c \cos t + d \sin t\). 2. We squared and summed both equations: \(x^2 + y^2 = (a \cos t + b \sin t)^2 + (c \cos t + d \sin t)^2\). 3. We expanded and simplified the equation. 4. We factored the simplified equation as \(x^2 + y^2 = (a^2 + c^2) \cos^2 t + (b^2 + d^2) \sin^2 t\). 5. We used the given conditions \(a^{2}+c^{2}=b^{2}+d^{2}=R^{2}\) and substituted them in our equation, obtaining \(x^2 + y^2 = R^2 \cos^2 t + R^2 \sin^2 t\). 6. We simplified the equation, reaching the standard equation of a circle: \(x^2 + y^2 = R^2\). Therefore, the given parametric equations describe a circle with radius R, centered at the origin, satisfying the provided conditions.

Step 1: Given Parametric Equations

We are given the parametric equations: $$x = a \cos t + b \sin t$$ $$y = c \cos t + d \sin t$$

Step 2: Square and Sum

Square both parametric equations and sum them: $$(x^2 = (a \cos t + b \sin t)^2)$$ $$(y^2 = (c \cos t + d \sin t)^2)$$ Adding both equations, $$x^2 + y^2 = (a \cos t + b \sin t)^2 + (c \cos t + d \sin t)^2$$

Step 3: Expand and Simplify

Expand the terms and simplify the equation, $$x^2 + y^2 = a^2 \cos^2 t + 2ab \cos t \sin t + b^2 \sin^2 t + c^2 \cos^2 t + 2cd \cos t \sin t + d^2 \sin^2 t$$ Utilize the given conditions: \(ab+cd=0\) can be written as \(2ab+2cd=0\), hence substitute \((2ab \cos t \sin t + 2cd \cos t \sin t)\) with \(0\) in above equation. $$x^2 + y^2 = a^2 \cos^2 t + b^2 \sin^2 t + c^2 \cos^2 t + d^2 \sin^2 t$$

Step 4: Factor the equation

Factor the simplified equation, $$x^2 + y^2 = (a^2 + c^2) \cos^2 t + (b^2 + d^2) \sin^2 t$$

Step 5: Utilize the Given conditions

We are given that \(a^{2}+c^{2}=b^{2}+d^{2}=R^{2}\), replace these in the equation, $$x^2 + y^2 = R^2 \cos^2 t + R^2 \sin^2 t$$

Step 6: Simplify and obtain the standard equation of circle

Factor and simplify the equation, $$x^2 + y^2 = R^2 (\cos^2 t + \sin^2 t)$$ Since, \(\cos^2 t + \sin^2 t = 1\), we get the standard equation of a circle, $$x^2 + y^2 = R^2$$ It has been shown that the given parametric equations describe a circle centered at the origin with radius R, satisfying the provided conditions.

Key Concepts

Circle

A circle is a perfect round shape where every point on its boundary is equidistant from a fixed point known as the center. This constant distance from the center is called the radius. In standard terms, the equation of a circle in the Cartesian plane with center at the origin is \( x^2 + y^2 = R^2 \), where \( R \) is the radius. The exercise provides a way to verify that certain parametric equations, which are equations that use one or more parameters to describe a curve, define a circle with radius \( R \). Rather than using the usual \( x \) and \( y \) coordinates directly, these parametric equations use a parameter, here represented by \( t \), and trigonometric functions to express the circle. This formulation easily captures motion around the circle, often useful in physics and engineering.

Trigonometric Functions

Trigonometric functions such as cosine (\(\cos\)) and sine (\(\sin\)) are pivotal in describing circles and periodic phenomena. These functions help establish the relationship between angles and sides in right-angled triangles and extend well into the unit circle definition. In our exercise, \(a \cos t + b \sin t\) and \(c \cos t + d \sin t\) utilize these trigonometric functions to describe coordinates \((x, y)\) on a circle parametrically.

• The sine function relates to the vertical position on the unit circle.
• The cosine function relates to the horizontal position on the unit circle.
• By varying \( t \), the angle parameter, the coordinates \((x, y)\) trace out a circle.

These functions are useful for modelling rotations and oscillations, making them indispensable tools in mathematical, scientific, and engineering applications.

Radius

The radius is a crucial aspect of a circle, defining its size. It is the linear distance from the center of the circle to any point on its perimeter. In the Introduction to Parametric Equations, the radius \( R \) is derived from conditions \( a^2 + c^2 = R^2 \) and \( b^2 + d^2 = R^2 \). This condition guarantees that as the parameter \( t \) varies, the points defined by the equations always lie on the circle defined by \( x^2 + y^2 = R^2 \).

• The circle’s radius determines the spatial scope that the circle covers.
• In our parametric equations, the equality conditions indicate that \( a, b, c, \) and \( d \) are aligned such that they form a specific radius \( R \) when squared and summed appropriately.

The concept of the radius is fundamental when graphing and solving circle-related problems, as it is a measure of the circle's scale and orientation centre.

Geometric Proof

Geometric proofs use logical reasoning to show why certain mathematical statements are true. In this context, the problem requires proving that the given parametric equations describe a circle with radius \( R \). We approach this proof by manipulating the given equations with basic algebraic operations until the expression \( x^2 + y^2 = R^2 \) is reached, illustrating that a circle is indeed defined by these parameters.

• Geometric proof often involves confirming that transformations maintain geometric properties like distance and angles.
• In the solution, expanding and simplifying using algebraic identities such as \((a+b)^2 = a^2 + 2ab + b^2\) helps isolate necessary conditions that describe a circle.
• Using the trigonometric identity \( \, \cos^2 t + \sin^2 t = 1 \, \) finalizes the proof by confirming that, given the constraints, the parametric equations indeed align with the standard circle equation \( x^2 + y^2 = R^2 \).

This approach in geometric proofs is routine in mathematics to establish the validity of conditions that define certain figures or concepts.