Problem 12 - Algebra and Trigonometry

Question

Test the polar equation for symmetry with respect to the polar axis, the pole, and the line \(\theta=\pi / 2 .\) $$ r=5 \cos \theta \csc \theta $$

Step-by-Step Solution

Answer

The polar equation is not symmetric with respect to the polar axis, the pole, or the line \(\theta = \frac{\pi}{2}\).

Step 1: Test for Symmetry with Respect to the Polar Axis

A polar equation is symmetric with respect to the polar axis if replacing \( \theta \) with \( -\theta \) yields an equivalent equation. Start by substituting \(-\theta\) into the given equation:\[ r = 5 \cos(-\theta) \csc(-\theta) \]Using the identities \(\cos(-\theta) = \cos(\theta)\) and \(\csc(-\theta) = -\csc(\theta)\), the equation becomes:\[ r = 5 (\cos(\theta))(-\csc(\theta)) = -5 \cos(\theta) \csc(\theta) \]This result is not equivalent to the original equation; thus, the equation is not symmetric with respect to the polar axis.

Step 2: Test for Symmetry with Respect to the Pole

A polar equation is symmetric with respect to the pole if replacing \( r \) by \(-r\) yields an equivalent equation. Replace \( r \) by \(-r \):\[-r = 5 \cos(\theta) \csc(\theta)\]Multiply through by \(-1\):\[r = -5 \cos(\theta) \csc(\theta)\]This is equivalent to the previous step for polar axis symmetry, and once again it is not the same as the original equation. Thus, the equation is not symmetric with respect to the pole.

Step 3: Test for Symmetry with Respect to the Line \(\theta = \frac{\pi}{2}\)

A polar equation is symmetric with respect to the line \(\theta = \frac{\pi}{2}\) if replacing \( \theta \) with \( \pi - \theta \) gives an equivalent equation. Substitute \( \pi - \theta \) for \( \theta \):\[ r = 5 \cos(\pi - \theta) \csc(\pi - \theta) \]Using the identities \(\cos(\pi - \theta) = -\cos(\theta)\) and \(\csc(\pi - \theta) = \csc(\theta)\), the equation becomes:\[ r = 5(-\cos(\theta)) \csc(\theta) = -5 \cos(\theta) \csc(\theta) \]This result is not the same as the original equation. Therefore, the equation is not symmetric with respect to the line \(\theta = \frac{\pi}{2}\).

Key Concepts

Symmetry in Polar Coordinates

Understanding symmetry in polar coordinates is essential when analyzing polar equations, which are often graphed on the polar coordinate system. Symmetry can help simplify these equations and provide insight into their graphs. Polar coordinates use the radius \(r\) and angle \(\theta\) to represent points in a plane.
There are several types of symmetry that can be useful to recognize:

• Symmetry with respect to the polar axis: This is similar to symmetry about the x-axis in Cartesian coordinates. If an equation remains unchanged after replacing \(\theta\) with \(-\theta\), it is symmetric with respect to the polar axis.
• Symmetry with respect to the pole (origin): This kind of symmetry indicates that the graph looks the same if duplicated across the pole. If replacing \(r\) with \(-r\) results in the original equation, symmetry exists regarding the pole.
• Symmetry with respect to the line \(\theta = \frac{\pi}{2}\): Similar to symmetry about the y-axis, this occurs when replacing \(\theta\) with \(\pi - \theta\) results in the original equation.

Recognizing these symmetries can greatly aid in sketching polar curves and understanding their behavior.

Trigonometric Identities

Trigonometric identities are crucial for solving and simplifying expressions in polar equations. These identities express relationships between different trigonometric functions, allowing us to transform and evaluate expressions.
Some basic trigonometric identities include:

• Reciprocal identities: \(\csc(\theta) = \frac{1}{\sin(\theta)}\), \(\sec(\theta) = \frac{1}{\cos(\theta)}\), \(\cot(\theta) = \frac{1}{\tan(\theta)}\).
• Even and odd functions: \(\cos(-\theta) = \cos(\theta)\), \(\sin(-\theta) = -\sin(\theta)\), \(\tan(-\theta) = -\tan(\theta)\).
• Co-function identities: \(\sin(\frac{\pi}{2} - \theta) = \cos(\theta)\), \(\cos(\frac{\pi}{2} - \theta) = \sin(\theta)\).

These identities can transform expressions into simpler forms or forms that reveal symmetries. For example, in the original problem, the identity \(\cos(\pi - \theta) = -\cos(\theta)\) alters \(5 \cos(\pi - \theta) \csc(\pi - \theta)\) to more clearly test for symmetry.

Testing Symmetry in Polar Equations

Testing polar equations for symmetry involves using transformations and trigonometric identities to determine if the equation remains unchanged under specific conditions.
To test symmetry systematically:

• Polar axis symmetry: Replace \(\theta\) with \(-\theta\) and simplify the equation. If the resulting expression is equivalent to the original equation, it has polar axis symmetry. In the exercise, replacing produced \(-5 \cos(\theta) \csc(\theta)\) versus the original \(5 \cos(\theta) \csc(\theta)\), indicating no symmetry.
• Pole symmetry: Substitute \(r\) with \(-r\). Simplification should reveal if substituting does not alter the original equation. If not equivalent, as with this equation, pole symmetry doesn't exist.
• Line \(\theta = \frac{\pi}{2}\) symmetry: Replace \(\theta\) with \(\pi - \theta\). A resulting expression equivalent to the original suggests this symmetry is present. The transformation\(-5 \cos(\theta) \csc(\theta)\) again signals a lack of symmetry.

Understanding these methods equips you to quickly determine symmetry, simplifying graphing and solution processes for polar equations.