Plot the cissoid r=2 sinθtanθand show that its equation in rectangular coordinates is y^2=(x^3)/

step 1 We want to plot the syssoid r equals to 2 sine teter, tangent teter. You can see from Desmos her

Plot the cissoid r=2 sinθtanθand show that its equation in...

Key Concepts

Trigonometric Identities

Trigonometric identities, such as tan? = sin?/cos?, are essential tools in simplifying and manipulating trigonometric expressions when converting equations. They allow for the rewriting of polar equations to align with the rectangular coordinate relationships, facilitating the transformation process.

Coordinate Transformation

Coordinate transformation involves converting equations between different coordinate systems, such as from polar to rectangular coordinates. This process requires the use of relationships like x = r cos? and y = r sin? to eliminate the polar variables in favor of x and y, making the equation easier to analyze or graph in a coordinate system with standard axes.

Curve Plotting

Curve plotting involves graphing an equation to visualize the geometric shape it represents. Understanding how a curve behaves—such as its symmetry, intersections, and asymptotic behavior—is aided through transforming its equation into a more suitable coordinate system or form for analysis.

Polar Coordinates

Polar coordinates describe a point's location in the plane by a distance from a fixed point (the pole) and an angle from a fixed direction, usually measured in radians. This system is especially useful for representing curves that exhibit radial symmetry or are more naturally defined in terms of angles and radii.

Rectangular Coordinates

Rectangular coordinates, or Cartesian coordinates, represent points in the plane using two perpendicular axes, typically labeled as x and y. Converting equations from polar to rectangular form allows one to analyze and graph the curves within the familiar Cartesian framework.

Transcript

00:01 We want to plot the syssoid r equals to 2 sine teter, tangent teter.

00:06 You can see from desmos here, i've plotted it from 0 to 2 pi for the teter.

00:11 Remember to put the bracket around sine teter and around tangent teter, so they are treated as two separate functions multiplied together.

00:22 So this is our sysmite.

00:24 Now over to this side, we want to show that the sysoid, which is r equals to 2 sine teter, tendon teter.

00:32 Can be expressed into its rectangular coordinate form.

00:37 Now, before we do that, let's take a look.

00:40 Of any point in the x, y space, or the cartesian system, you know that this part here is x and this part here is y.

00:49 Now, the point can be joined to the origin, with this distance being r, and the angle that makes with the positive excesses anti -clockwise.

01:02 Let's call it theta.

01:05 So in polar coordinate, you can express it in terms of r and theta.

01:09 Now do take note this distance here to origin is x, and this vertical distance to origin is to the x or 6 is y.

01:17 Now by pydecoris theorem, we know that r squared is equal to x squared plus y square.

01:23 So r is square root x square plus y square.

01:28 By trigon, we know that x over r is cosine tether.

01:33 And we also know that y over r is sine theta...

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