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Chapter 1: Problem 151

Determine whether the graphs of the polar equation are symmetric with respect to the \(x\)-axis, the \(y\) -axis, or the origin. $$ r=\cos \left(\frac{\theta}{5}\right) $$

Short Answer

Expert verified
The graph is symmetric with respect to the x-axis and the origin.

Step by step solution

01

Test for Symmetry with respect to the x-axis

To test if a polar equation is symmetric with respect to the x-axis, substitute \(\theta\) with \(-\theta\). For the given equation, this yields:\[r = \cos \left(\frac{-\theta}{5}\right) = \cos \left(-\frac{\theta}{5}\right)\]Since the cosine function is even, \(\cos\left(-x\right) = \cos\left(x\right)\), so:\[\cos \left(-\frac{\theta}{5}\right) = \cos \left(\frac{\theta}{5}\right)\]Thus, \(r = \cos \left(\frac{\theta}{5}\right)\) confirms that the graph is symmetric with respect to the x-axis.
02

Test for Symmetry with respect to the y-axis

To check symmetry with respect to the y-axis, replace \(\theta\) with \(\pi - \theta\). Substitute into the equation:\[r = \cos \left(\frac{\pi - \theta}{5}\right)\]Applying the identity for cosine, \(\cos(\pi - x) = -\cos(x)\), we get:\[r = -\cos \left(\frac{\theta}{5}\right)\]This result, \(r = -\cos\left(\frac{\theta}{5}\right)\), is not equal to the original equation, so the graph is not symmetric with respect to the y-axis.
03

Test for Symmetry with respect to the origin

To test for origin symmetry, replace \(r\) with \(-r\) and \(\theta\) with \(\theta + \pi\). Substitute in the equation:\[-r = \cos \left(\frac{\theta + \pi}{5}\right)\]Using the cosine angle addition formula, \(\cos(x + \pi) = -\cos(x)\), gives:\[-r = -\cos \left(\frac{\theta}{5}\right)\]This simplifies to \(r = \cos \left(\frac{\theta}{5}\right)\), which matches the original equation, confirming symmetry with respect to the origin.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Polar Coordinates
Polar coordinates are a different way of representing points on a plane, as opposed to the usual Cartesian coordinates, which use the x and y-axis. Here, each point is defined by a distance "r" from the origin and an angle "\(\theta\)" measured from the positive x-axis. This means:
  • "r" tells you how far you are from the center, the pole.
  • "\(\theta\)" shows the direction or angle you need to move to reach the point.
This can be very useful for certain types of graphs, especially those that have circular or spiral patterns. In polar coordinates, a point can also have many equivalent forms by varying \(\theta\) because angles are periodic. This brings a lot of utility in problems involving symmetry, making it easier to identify symmetries with respect to different axes or the origin.
Trigonometric Symmetry
Understanding symmetry is crucial in identifying how a function behaves when transformed. In polar graphs, this involves checking whether the graph remains unchanged when you rotate it or flip it. Symmetry tests include:
  • Symmetry with respect to the x-axis: Substitute \(\theta\) with \(-\theta\). If the equation remains unchanged, the graph is x-axis symmetric.
  • Symmetry with respect to the y-axis: Replace \(\theta\) with \(\pi - \theta\). If the equation is the same, the graph is y-axis symmetric.
  • Symmetry with respect to the origin: Replace \(r\) with \(-r\) and \(\theta\) with \(\theta + \pi\). If unchanged, it鈥檚 origin symmetric.
These transformations allow you to test symmetries based on trigonometric identities like \(\cos(-\theta) = \cos(\theta)\) (even function) and \(\cos(\pi - \theta) = -\cos(\theta)\). Recognizing these symmetries helps in visualizing the graph more easily.
Cosine Function
The cosine function, often denoted as \(\cos(\theta)\), is one of the fundamental trigonometric functions. It's periodic with a period of \(2\pi\), meaning it repeats its values every \(2\pi\) radians (or 360 degrees). Some key characteristics include:
  • Even Function: This means that \(\cos(-x) = \cos(x)\), making symmetry tests simpler.
  • Range: The values are strictly between -1 and 1. This makes it ideal for representing amplitudes in wave functions.
  • Graph: A smooth wave that starts from 1 at \(\theta=0\), decreases to -1, and returns to 1 at \(2\pi\).
Within polar equations like \(r = \cos(\frac{\theta}{5})\), the cosine function scales and shifts based on the angle division, affecting the spiral and circular symmetry within the graph. Mastering cosine allows for recognizing patterns and symmetries more intuitively in polar plots.

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Most popular questions from this chapter

Convert the parametric equations of a curve into rectangular form. No sketch is necessary. State the domain of the rectangular form. $$ x=2 \cosh t, \quad y=4 \sinh t $$

Find \(d^{2} y / d x^{2}\). $$ x=e^{-t}, \quad y=t e^{2 t} $$

Find the arc length of the curve on the indicated interval of the parameter. $$ x=1+t^{2}, \quad y=(1+t)^{3}, \quad 0 \leq t \leq 1 $$

Find the area enclosed by the ellipse \(x=a \cos \theta, y=b \sin \theta, 0 \leq \theta<2 \pi\).

Sketch the curve known as an epitrochoid, which gives the path of a point on a circle of radius \(b\) as it rolls on the outside of a circle of radius \(a\). The equations are $$ \begin{array}{l} x=(a+b) \cos t-c \cdot \cos \left[\frac{(a+b) t}{b}\right] \\ y=(a+b) \sin t-c \cdot \sin \left[\frac{(a+b) t}{b}\right] \end{array} $$ Let \(a=1, b=2, c=1\).
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