Test for symmetry with respect to Q=pi/2 , the polar axis and the pole.
r = 16 cos3Q
Please explain. I do not know how to do this. Thank You!!
if Q = pi/2 + h
r = 16 cos (3pi/2 +3h)
= 16[ cos 3pi/2 cos 3h - sin 3pi/2 sin h ]
cos 3pi/2 = 0
sin 3pi/2 = -1
so
16 sin h
is it the same for -h?
if Q = pi/2 - h
r = 16 cos (3 pi/2 -3h)
=16[ cos 3pi/2 cos 3h + sin3 pi/2 sin 3h
=16 (-sin h)
sin h is not the same as -sin h
so no, not symmetric about 3pi/2
To test for symmetry with respect to a given line or point in polar coordinates, we need to examine the equation and see if it remains unchanged after applying certain transformations.
In this case, the equation is given by r = 16cos^3(Q).
1. Symmetry with respect to Q = π/2 (the vertical line):
- Replace Q in the equation with (π/2 - Q). If the equation remains the same, then it has symmetry with respect to the line Q = π/2.
Let's substitute (π/2 - Q) for Q in the equation:
r = 16cos^3((π/2 - Q))
To simplify further, we can use a trigonometric identity: cos^3(α) = sin^3(π/2 - α)
r = 16sin^3(Q)
Comparing this with the original equation, we can see that they are not the same. Therefore, there is no symmetry with respect to Q = π/2 (the vertical line).
2. Symmetry with respect to the polar axis (horizontal line):
- Replace r with -r in the equation. If the equation remains the same, then it has symmetry with respect to the polar axis.
Let's substitute -r for r in the equation:
-r = 16cos^3(Q)
Rearranging the equation, we have:
r = -16cos^3(Q)
Comparing this with the original equation, we can see that they are not the same. Therefore, there is no symmetry with respect to the polar axis.
3. Symmetry with respect to the pole (the origin):
- Replace Q with (π - Q) in the equation. If the equation remains the same, then it has symmetry with respect to the pole.
Let's substitute (π - Q) for Q in the equation:
r = 16cos^3(π - Q)
To simplify further, we can use a trigonometric identity: cos(π - α) = -cos(α)
r = -16cos^3(Q)
Comparing this with the original equation, we can see that they are not the same. Therefore, there is no symmetry with respect to the pole.
In conclusion, the given polar equation r = 16cos^3(Q) does not exhibit symmetry with respect to Q = π/2, the polar axis, or the pole.