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### Course: Algebra (all content)>Unit 14

Lesson 18: Parametric equations

# Removing the parameter in parametric equations (example 2)

Sal is given x=3cost and y=2sint and he finds an equation that gives the relationship between x and y (spoiler: it's an ellipse!). Created by Sal Khan.

## Want to join the conversation?

• Can anyone explain the idea of "arc sine" in a little more detail? My teachers have always said sine inverse. I understood what Sal was saying around , but what is arcsin and where does it come from? Any information is appreciated, thanks. :)
• The point that he's kinda meandering around is that arcsin and inverse sine are just different names (and notations) for the same operation.

The term "inverse sine" makes more sense to mathematicians, but "arcsin" might be clearer to the old timey people who actually used it. For instance, draw a sector of a circle with unit radius with one radius as the "base" and draw the "altitude" from the base to the third point of the "triangle". If you knew that altitude was x and wanted to calculate the angular measure of the arc of the triangle, then in that sense it makes sense that you could define that calculation as the arc sine of x, which is exactly the same as what you think of as the inverse sine.
• Does it make a difference if the trig term does not have the same theta term with it? For example, if we are given x= sin(theta) and y=cos(2theta) can we follow this example of converting to x and y (if so, how would that work out?)

I'm working on this particular question and I went with the double angle identity of cos, so cos(2theta)=cos(theta)*cos(theta)-sin(theta)*sin(theta) which is simply cos^2(theta)-sin^2(theta). But then how do I relate that to the x=sin(theta)?

Any help is appreciated. Thanks!
Javier
• Theta is just a variable that is often used for angles, it's interchangeable with x.
(1 vote)
• Why arcsin y and 1/sin y is not the same thing ?
(1 vote)
• Inverse of a function is not Multiplicative Inverse

arcsiny is the Inverse of the function siny and 1/siny is the multiplicative inverse of siny. They can't the same thing.
Also the Domain and the range are different for these two functions. So, there's no way for them to be the same.
• Where did Sal get cos^2t+sin^2t=1? Is that a trig. identity? Why?
(1 vote)
• Wait, so ((sin^-1)(y)) = arcsin(y) not 1/sin(y),
but ((sin^-2)(y)) does = 1/(sin(y))^2
??
• it is very confusing, which is why Sal prefers to use arcsin instead of sin^-1
• Instead of cos and sin, what happens if it was tangent instead? How would it be solved?
For example, if x = tan^2(t) and y = tan(t), it wouldn't make an ellipse would it?
• Is the graph of an ellipse a function?
(1 vote)
• in when Sal uses pi and pi/2 for his time, is there a reason why he couldn't use 90 or 180?
(1 vote)
• Khan is using angles in radians probably because it is more intuitive rather than in degrees.
cos(90º) = 0 = cos(Pi/2) Both expressions are equivalent (just mind the angle units)
• If the parameter were 0 < t < 3pi/2 , how do you remove the parameter?
• why it's necessary to place + - before square-root?
(1 vote)
• when you square a number, the answer is always positive no matter if the original number is negative.
So if you have (x)^2, it is always x^2. Also if you have (-x)^2, it is also equal to x^2

Example (Try in a calculator if you want, it is always positive):
(5)^2=25
also
(-5)^2=25

So when you do any square root, you have to put+- just to be clear to anyone including yourself that the answer might also be negative.
You see where this is important in quadratic equations a lot when you are doing the quadratic formula to find factors especially.
Hope this helps.