An equation of a curve in terms of polar coordinates r and ? is called a Polar Equation.
The number of petals in the graph of a polar equation r = a sin nθ or r = a cos nθ is 'n', if n is odd, and '2n', if n is even, where 'a' is constant.
If the polar equation is r(- θ) = r(θ), then the curve is symmetrical about the horizontal axis.
If the polar equation is r(π - θ) = r(θ), then the curve is symmetrical about the vertical axis.
r = 3sin 4θ and r = 2cos 5θ are polar equations as they are written in terms of r and θ.
A. r = tan2 θ sec θ
B. r = tan2 θ
C. r 2 = tan2 θ sec θ
D. r = cot2 θ cos θ
Correct Answer: A
Step 1: y2 = x3 [Rectangular equation.]
Step 2: (rsin θ)2 = (rcos θ)3 [Use y = rsin θ and x = rcos θ]
Step 3: r2sin2θ = r3cos3 θ
Step 4: sin2 θ / cos3 θ = r3/ r2
Step 5: r = tan2 θ sec θ [Use trigonometric definitions.]
Step 6: So, the polar equation is r = tan2 θ sec θ.
Q1: Which of the following is the polar equation of the line y = x?
Q2: The graph of r = a cos(nθ) has how many petals when n is even?
Q3: Which test determines symmetry about the x-axis in a polar equation?
Q: How do I convert a rectangular equation to a polar equation?
A: Use the substitutions x = r cos θ and y = r sin θ and simplify the resulting equation.
Q: How do I determine the symmetry of a polar curve?
A: Check if the equation remains unchanged when θ is replaced by -θ (horizontal symmetry), π - θ (vertical symmetry), or -r, θ + π (symmetry about the pole).
Q: What does a negative value of 'r' mean?
A: A negative 'r' indicates a point located in the opposite direction of the angle θ from the origin.