Parametric equations with sine and cosine

TL;DR

The video explains how to convert parametric equations into Cartesian equations and shows the graphs of various parametric equations.

Transcript

you and let's talk about the first one right here Exodus cosine of T and y sine of T we're way out you see that X is the cosine and why it's the sign that reminds you what the unit circle isn't it okay you pretty much know that's the unit circle and not the way to do it is the following well if you just add these two up you don't know what X plus y... Read More

Key Insights

• 🙃 Parametric equations can be converted into Cartesian equations by squaring both sides.
• 🐎 Changing the coefficient of the angle in parametric equations affects the speed of traversal on the unit circle.
• ✖️ Multiplying the y-coordinate by a constant stretches the graph vertically.

Questions & Answers

Q: How do you convert parametric equations into Cartesian equations?

To convert parametric equations into Cartesian equations, you square both sides of the equations and simplify. This allows you to eliminate the parameter and obtain equations in terms of x and y.

Q: How does multiplying the y-coordinate by 2 affect the graph?

Multiplying the y-coordinate by 2 stretches the graph vertically. This results in an ellipse shape where the y-intercepts are at 2 and -2.

Q: What happens when the angles in the parametric equations have different values?

When the angles in the parametric equations are different, you cannot directly square both sides. Instead, you can use trigonometric identities to express one variable in terms of the other and then square both sides.

Q: How does changing the coefficient of the angle affect the graph?

Changing the coefficient of the angle affects the speed at which the graph is traced. A larger coefficient leads to a faster traversal of the unit circle, resulting in multiple loops in the same time period.

Summary & Key Takeaways

• The video introduces the concept of parametric equations and the unit circle.

• It demonstrates how to convert parametric equations into Cartesian equations by squaring both sides.

• The video shows the graphs of parametric equations with different coefficients, including circles, ellipses, and bowtie shapes.