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Magazine Problem 65
In Exercises 45-68, graph each equation. In Exercises 63-68, convert the equation from polar to rectangular form first and identify the resulting equation as a line, parabola, or circle. $$ r^{2} \cos ^{2} \theta-2 r \cos \theta+r^{2} \sin ^{2} \theta=8 $$
Problem 65
Consider the parametric equations \(x=a \sin t-\sin (a t)\) and \(y=a \cos t+\cos (a t)\). Use a graphing utility to explore the graphs for \(a=2,3\), and 4 .
Problem 66
$$ \text { For Exercises 65-76, simplify and express in standard form. } $$ $$ i^{99} $$
Problem 66
Consider the parametric equations \(x=a \sin t-\sin (a t)\) and \(y=a \cos t+\cos (a t)\). Use a graphing utility to explore the graphs for \(a=2,3\), and 4 .
Problem 66
In Exercises 61-72, use a calculator to express each complex number in rectangular form. $$ -5\left(\cos 320^{\circ}+i \sin 320^{\circ}\right) $$
Problem 66
In Exercises 63-74, find all complex solutions to the given equations. $$ x^{3}+1=0 $$
Problem 66
In Exercises 45-68, graph each equation. In Exercises 63-68, convert the equation from polar to rectangular form first and identify the resulting equation as a line, parabola, or circle. $$ r^{2} \cos ^{2} \theta-r \sin \theta=-2 $$
Problem 67
Consider the parametric equations \(x=\cos (a t)\) and \(y=\sin (b t)\). Use a graphing utility to explore the graphs for \(a=2\) and \(b=4, a=4\) and \(b=2, a=1\) and \(b=3\), and \(a=3\) and \(b=1\). Find the \(t\)-interval that gives one cycle of the curve.
Problem 67
In Exercises 61-72, use a calculator to express each complex number in rectangular form. $$ 3\left[\cos \left(\frac{11 \pi}{12}\right)+i \sin \left(\frac{11 \pi}{12}\right)\right] $$
Problem 67
In Exercises 63-74, find all complex solutions to the given equations. $$ x^{4}+16=0 $$
