# How do you sketch the graph of the polar equation and find the tangents at the pole of #r=2cos3theta#?

graph{(x^2+y^2)^2-8x^3+6x(x^2+y^2)=0 [-5, 5, -2.5, 2.5]}

loops are created.

Slope of the tangent

#=(-6sin 3theta sin theta +2 cos 3theta cos theta)

direction twice.

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To sketch the graph of the polar equation ( r = 2\cos(3\theta) ) and find the tangents at the pole, follow these steps:

**Plot Key Points**: Substitute various values of ( \theta ) to obtain corresponding values of ( r ). Plot these points on a polar coordinate system.**Symmetry Analysis**: Determine any symmetrical properties of the graph, such as symmetry about the polar axis (( \theta = 0 )) or the line ( \theta = \frac{\pi}{3} ).**Periodicity**: Consider the periodicity of the cosine function, which is ( 2\pi/3 ) in this case. This means the graph repeats every ( \frac{\pi}{3} ) radians.**Graph Construction**: Connect the plotted points smoothly, following the periodicity and any symmetrical properties.**Finding Tangents at the Pole**: To find the tangent at the pole, calculate the derivative of ( r ) with respect to ( \theta ), ( \frac{dr}{d\theta} ), and evaluate it when ( \theta = 0 ). The tangent at the pole will have an angle of inclination equal to ( \theta = 0 ) and a length equal to the value of ( r ) when ( \theta = 0 ).

Using these steps, you can sketch the graph of the polar equation ( r = 2\cos(3\theta) ) and find the tangents at the pole.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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