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Symmetry tests

A polar equation    describes a curve on the polar grid. The graph of a polar equation can be evaluated for three types of symmetry, as shown in [link] .

3 graphs side by side. (A) shows a ray extending into Q 1 and its symmetric version in Q 2. (B) shows a ray extending into Q 1 and its symmetric version in Q 4. (C) shows a ray extending into Q 1 and its symmetric version in Q 3. See caption for more information.
(a) A graph is symmetric with respect to the line θ = π 2 ( y -axis) if replacing ( r , θ ) with ( r , θ ) yields an equivalent equation. (b) A graph is symmetric with respect to the polar axis ( x -axis) if replacing ( r , θ ) with ( r , θ ) or ( r , π− θ ) yields an equivalent equation. (c) A graph is symmetric with respect to the pole (origin) if replacing ( r , θ ) with ( r , θ ) yields an equivalent equation.

Given a polar equation, test for symmetry.

  1. Substitute the appropriate combination of components for ( r , θ ) : ( r , θ ) for θ = π 2 symmetry; ( r , θ ) for polar axis symmetry; and ( r , θ ) for symmetry with respect to the pole.
  2. If the resulting equations are equivalent in one or more of the tests, the graph produces the expected symmetry.

Testing a polar equation for symmetry

Test the equation r = 2 sin θ for symmetry.

Test for each of the three types of symmetry.

1) Replacing ( r , θ ) with ( r , θ ) yields the same result. Thus, the graph is symmetric with respect to the line θ = π 2 . r = 2 sin ( θ ) r = −2 sin θ Even-odd identity r = 2 sin θ Multiply by −1 Passed
2) Replacing θ with θ does not yield the same equation. Therefore, the graph fails the test and may or may not be symmetric with respect to the polar axis. r = 2 sin ( θ ) r = −2 sin θ Even-odd identity r = −2 sin θ 2 sin θ Failed
3) Replacing r with r changes the equation and fails the test. The graph may or may not be symmetric with respect to the pole. r = 2 sin θ    r = −2 sin θ 2 sin θ Failed
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Test the equation for symmetry: r = 2 cos θ .

The equation fails the symmetry test with respect to the line θ = π 2 and with respect to the pole. It passes the polar axis symmetry test.

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Graphing polar equations by plotting points

To graph in the rectangular coordinate system we construct a table of x and y values. To graph in the polar coordinate system we construct a table of θ and r values. We enter values of θ into a polar equation    and calculate r . However, using the properties of symmetry and finding key values of θ and r means fewer calculations will be needed.

Finding zeros and maxima

To find the zeros of a polar equation, we solve for the values of θ that result in r = 0. Recall that, to find the zeros of polynomial functions, we set the equation equal to zero and then solve for x . We use the same process for polar equations. Set r = 0 , and solve for θ .

For many of the forms we will encounter, the maximum value of a polar equation is found by substituting those values of θ into the equation that result in the maximum value of the trigonometric functions. Consider r = 5 cos θ ; the maximum distance between the curve and the pole is 5 units. The maximum value of the cosine function is 1 when θ = 0 , so our polar equation is 5 cos θ , and the value θ = 0 will yield the maximum | r | .

Similarly, the maximum value of the sine function is 1 when θ = π 2 , and if our polar equation is r = 5 sin θ , the value θ = π 2 will yield the maximum | r | . We may find additional information by calculating values of r when θ = 0. These points would be polar axis intercepts, which may be helpful in drawing the graph and identifying the curve of a polar equation.

Questions & Answers

root under 3-root under 2 by 5 y square
Himanshu Reply
The sum of the first n terms of a certain series is 2^n-1, Show that , this series is Geometric and Find the formula of the n^th
amani Reply
cosA\1+sinA=secA-tanA
Aasik Reply
why two x + seven is equal to nineteen.
Kingsley Reply
The numbers cannot be combined with the x
Othman
2x + 7 =19
humberto
2x +7=19. 2x=19 - 7 2x=12 x=6
Yvonne
because x is 6
SAIDI
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simplify each radical by removing as many factors as possible (a) √75
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Karl Reply
what is the value of x in 4x-2+3
Vishal Reply
give the complete question
Shanky
4x=3-2 4x=1 x=1+4 x=5 5x
Olaiya
hi can you give another equation I'd like to solve it
Daniel
what is the value of x in 4x-2+3
Olaiya
if 4x-2+3 = 0 then 4x = 2-3 4x = -1 x = -(1÷4) is the answer.
Jacob
4x-2+3 4x=-3+2 4×=-1 4×/4=-1/4
LUTHO
then x=-1/4
LUTHO
4x-2+3 4x=-3+2 4x=-1 4x÷4=-1÷4 x=-1÷4
LUTHO
A research student is working with a culture of bacteria that doubles in size every twenty minutes. The initial population count was  1350  bacteria. Rounding to five significant digits, write an exponential equation representing this situation. To the nearest whole number, what is the population size after  3  hours?
David Reply
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Haidar Reply
Need help with math
Peya
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litshani
( cosec Q _ cot Q ) whole spuare = 1_cosQ / 1+cosQ
Aarav Reply
A guy wire for a suspension bridge runs from the ground diagonally to the top of the closest pylon to make a triangle. We can use the Pythagorean Theorem to find the length of guy wire needed. The square of the distance between the wire on the ground and the pylon on the ground is 90,000 feet. The square of the height of the pylon is 160,000 feet. So, the length of the guy wire can be found by evaluating √(90000+160000). What is the length of the guy wire?
Maxwell Reply
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Practice Key Terms 9

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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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