What is the name of the shape graphed by the function r^2=9 cos(2 theta)?

The name of the shape graphed by the function r ^ 2 = 9 cos (2 theta) is called the “lemniscate”. A lemniscate is a plane curve with a feature shape which consists of two loops that meet at a central point. The curve is also sometimes called as the lemniscate of Bernoulli.

Explanation:

The period of coskθ is 2π/k. In this case, k = 2 therefore the period is π.

r ^ 2 = 9 cos 2θ ≥0 → cos 2θ ≥0. So easily one period can be chosen as θ ∈ [0, π] wherein cos 2θ ≥0.

As cos(2(−θ)) = cos2θ, the graph is symmetrical about the initial line.

Also, as cos (2(pi-theta) = cos 2theta, the graph is symmetrical about the vertical θ = π/2

A Table for half period [0,π4/] is adequate for the shape in Quarter1

Use symmetry for the other three quarters:

(r, θ) : (0,3)(3/√√2,π/8)(3√2/2,π/6)(0,π/4)

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step-by-step explanation:

step-by-step explanation:

you divide the denominator by the numerator on both sides and if the quotients are equal then the fractions are equivalent