Problem 64 - Precalculus

Question

Show that the equation $$x^{2}+y^{2}=r^{2}$$ is invariant under rotation of axes.

Step-by-Step Solution

Answer

After rotating the axes and simplifying the equation, the equation remains in the form \(x^{2}+y^{2}=r^{2}\). Therefore, this equation is indeed invariant under rotation of axes.

Step 1: Write down the rotation formula

Let's recall the formula for counter-clockwise rotation of axes by \( \theta \) degrees: \(x = x' \cos(\theta) - y' \sin(\theta)\) and \(y = x'\sin(\theta) + y'\cos(\theta)\). In these equations, \((x,y)\) are original coordinates and \((x',y')\) are coordinates after rotation.

Step 2: Substitute the rotation formulae into the given equation

Next, substitute the rotation formulae into the given equation \(x^{2}+y^{2}=r^{2}\) to obtain \((x' \cos(\theta) - y' \sin(\theta))^{2} + (x'\sin(\theta) + y'\cos(\theta))^{2} = r^2\).

Step 3: Simplify the equation

Simplify this equation using trigonometric identities. The equation becomes \((x')^{2}\cos^{2}(\theta) + (y')^{2}\sin^{2}(\theta) - 2x'y'\sin(\theta)\cos(\theta) + (x')^{2}\sin^{2}(\theta) + (y')^{2}\cos^{2}(\theta) + 2x'y'\sin(\theta)\cos(\theta) = r^{2}\). This can be further simplified by collecting like terms and using the Pythagorean identity \(\cos^{2}(\theta)+\sin^{2}(\theta)=1\). The simplified equation becomes \((x')^{2} + (y')^{2} = r^{2}\).

Step 4: Compare the resulting equation with the given equation

Compare the resulting equation \((x')^{2} + (y')^{2} = r^{2}\) with the original equation \(x^{2}+y^{2}=r^{2}\). Notice that the resulting equation has the same form as the original equation.

Key Concepts

Invariant Equations

An invariant equation remains unchanged even when transformations, like rotations, are applied to the system of coordinates. This concept is crucial in understanding physical laws and geometric problems that possess symmetry.

Let’s illustrate this with the circle equation, \( x^2 + y^2 = r^2 \). Here, \( r \) is the radius. When you rotate the axes around the origin by an angle \( \theta \), you expect this equation to still represent the same circle.

• The original coordinates \((x, y)\) change to the new coordinates \((x', y')\), but the equation maintains its form.
• This property of remaining unchanged indicates the equation's rotational invariance.

Such equations are vital in ensuring that physical phenomena appear the same, irrespective of how you view them from different angles. Thus, an invariant equation like this confirm consistency across transformations.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for every value of the variable. They help simplify complex problems, especially those involving angles and periodic geometry.

In the problem context, after applying the rotation formula, identities like \( \cos^2(\theta) + \sin^2(\theta) = 1 \) become handy. This specific identity is known as the Pythagorean identity.

• It helps to combine and simplify terms in the equation obtained after substitution during coordinate transformations.
• These identities reduce the complexity and make an otherwise difficult algebraic procedure manageable.

Such simplifications allow us to focus on the pivotal concept without getting lost in the math, maintaining the essence while rotating the coordinate axes.

Coordinate Transformation

Coordinate transformation involves changing the positions of points described in a particular system to a new reference frame. A classic example is the rotation of axes, where we rotate the entire system based on a central point, typically the origin.

For a counter-clockwise rotation by \(\theta\) degrees, the transformation equations are:\[\begin{align*} x &= x' \cos(\theta) - y' \sin(\theta) \ y &= x' \sin(\theta) + y' \cos(\theta)\end{align*}\]These equations show how each point in the original system \((x,y)\) can be re-expressed in the rotated system \((x',y')\).

• Coordinate transformations are essential in physics and engineering for analyzing systems from different perspectives.
• They simplify problems by aligning coordinate systems with symmetry, thereby easing calculations.

Understanding coordinate transformations helps in visualizing how geometric shapes or physical phenomena behave under different orientations, enhancing both comprehension and problem-solving skills.