Problem 10 - Algebra and Trigonometry

Question

Determine the equation of the given conic in \(X Y\) -coordinates when the coordinate axes are rotated through the indicated angle. $$x^{2}+2 y^{2}=16, \quad \phi=\sin ^{-1} \frac{3}{5}$$

Step-by-Step Solution

Answer

The rotated conic equation is \( X^2 - XY + Y^2 = 10 \).

Step 1: Determine the Rotation Angle

Determine the rotation angle \( \phi \) given by \( \phi = \sin^{-1} \frac{3}{5} \). Find \( \cos \phi \) using the Pythagorean identity. If \( \sin \phi = \frac{3}{5} \), then \( \cos \phi = \sqrt{1 - \sin^2 \phi} = \sqrt{1 - \left(\frac{3}{5}\right)^2} = \frac{4}{5} \).

Step 2: Write Down Rotation Formulas

Use the rotation formulas to find the new coordinates: \( X = x \cos \phi + y \sin \phi \) and \( Y = -x \sin \phi + y \cos \phi \). Substitute \( \sin \phi = \frac{3}{5} \) and \( \cos \phi = \frac{4}{5} \) into the formulas: \( X = \frac{4}{5}x + \frac{3}{5}y \) and \( Y = -\frac{3}{5}x + \frac{4}{5}y \).

Step 3: Substitute and Simplify

Substitute \( x = \frac{4}{5}X - \frac{3}{5}Y \) and \( y = \frac{3}{5}X + \frac{4}{5}Y \) into the original equation \( x^2 + 2y^2 = 16 \). Expand and simplify the expression to find the equation in terms of \( X \) and \( Y \).

Step 4: Calculate Resultant Equation

Perform the multiplication, expansion, and collect like terms. After substitution, you obtain: \( \frac{16}{25}X^2 - \frac{24}{25}XY + \frac{9}{25}Y^2 + 2\left(\frac{9}{25}X^2 + \frac{24}{25}XY + \frac{16}{25}Y^2\right) = 16 \). Simplify to get the final equation: \( X^2 - XY + Y^2 = 10 \).

Key Concepts

Rotation of Axes

When you rotate the axes in geometry, you are essentially changing the perspective from which you view a figure. This helps simplify equations if a rotation makes the original axes align better with the shape of interest.

To rotate the axes, you'll need to decide on a rotation angle, which is often given using trigonometric functions. In our example, we use \( \phi = \sin^{-1} \frac{3}{5} \), leading us to calculate \( \cos \phi \) using the identity \( \cos \phi = \sqrt{1 - \sin^2 \phi} \).

The rotation transforms the coordinates \( (x, y) \) to new coordinates \( (X, Y) \):

• \( X = x \cos \phi + y \sin \phi \)
• \( Y = -x \sin \phi + y \cos \phi \)

This transformation allows us to express equations more simply in these new rotated axes.

Equation Transformation

Transforming an equation involves substituting variables according to a new coordinate system. This helps simplify complex problems by changing the perspective of the equation. In the exercise, we substitute \( x \) and \( y \) into their expressions in terms of \( X \) and \( Y \):

• \( x = \frac{4}{5}X - \frac{3}{5}Y \)
• \( y = \frac{3}{5}X + \frac{4}{5}Y \)

After substitution into the equation \( x^2 + 2y^2 = 16 \), it becomes crucial to perform algebraic operations carefully:

1. Expand all terms.2. Collect like terms.3. Simplify the expression.

Following these steps results in our transformed equation: \( X^2 - XY + Y^2 = 10 \). This means the original ellipse now has an equation aligned with the new axes.

Trigonometric Identities

Trigonometric identities are key in calculations involving angles and rotations. These identities, like the Pythagorean identity, help find unknown trigonometric values using known ones.

Consider when \( \sin \phi = \frac{3}{5} \). To find \( \cos \phi \), you use the Pythagorean identity:

• \( \cos \phi = \sqrt{1 - \sin^2 \phi} = \sqrt{1 - \left(\frac{3}{5}\right)^2} = \frac{4}{5} \)

These values are then used in rotation formulas. They ensure accurate transformation of the coordinate system.

Trigonometric functions play a prominent role in geometry, especially when dealing with rotations. They relate angles to ratios of right triangle sides, effectively bridging linear and angular measurements.