Problem 37 - Algebra and Trigonometry

Question

Test for symmetry and then graph each polar equation. $$r=\sin \theta+\cos \theta$$

Step-by-Step Solution

Answer

The polar equation \(r= \sin \theta + \cos \theta\) is neither symmetric about the polar axis, nor the origin, nor the line \( \theta=\pi/2 \). Its graph is a straight line passing through the pole and making an angle \( \theta=\frac{\pi}{4} \) with the polar axis.

Step 1: Test Symmetry about the Polar Axis

This is done by replacing \( \theta \) with \( -\theta \) in the equation. If the equation remains unchanged, it is symmetric about the polar axis. So, \Substitute \( \theta = -\theta \) into the original equation to get \( r = \sin(-\theta) + \cos(-\theta) \). Using the properties of sine and cosine for negative angles, we rewrite this as \( r = -\sin(\theta) - \cos(\theta) \). Since this differs from the original equation, the graph of this equation is not symmetric about the polar axis.

Step 2: Test Symmetry about the Origin

This is done by replacing \( r \) with \( -r \) in the equation. If the equation remains unchanged, it is symmetric about the origin. But here, replacing \( r \) with \( -r \) does not produce the original equation, so it shows that the graph of the equation is not symmetric about the origin.

Step 3: Test Symmetry about the Line \( \theta=\pi/2 \)

This is done by replacing \( \theta \) with \( \pi - \theta \) in the equation. If the equation remains unchanged, it is symmetric about the line \( \theta=\pi/2 \). But upon substitution, the equation is not unaffected, therefore, it's not symmetric under this condition

Step 4: Convert Polar Equation to Rectangular Coordinates

Remember that in polar coordinates, \( x=r\cos \theta \) and \( y = r\sin \theta \). So the equation \( r=\sin \theta + \cos \theta \) can be rewritten as \( r = \frac{y}{r} + \frac{x}{r} \) which simplifies to \( x + y = r \). At this stage, the equation is in a simple and understandable format that can be plotted easily.

Step 5: Graph the Equation

The rectangular form of the equation can be graphed using basic knowledge of linear equations. We know this is a straight line with an x-intercept and a y-intercept at r. Identify at least two points and draw the line. You will notice that it's a straight line passing through the pole (origin) making an angle \( \theta=\frac{\pi}{4} \) with the polar axis.