(2012 paper) Q16 (b) (i) (1 Viewer)

[Hypem]

http://www.boardofstudies.nsw.edu.au/hsc_exams/hsc2012exams/pdf_doc/2012-hsc-exam-maths.pdf

This is probably really obvious, but:

How do you find the coordinates of T?

 

[Carrotsticks]

 

[Hypem]

I still don't really understand what a parametric coordinate is? I've never learnt that.

 

[Hypem]

Oh wait, I get it. I just had to think of projectile motion from Physics to understand haha!

 

[Sy123]

http://www.boardofstudies.nsw.edu.au/hsc_exams/hsc2012exams/pdf_doc/2012-hsc-exam-maths.pdf

This is probably really obvious, but:

How do you find the coordinates of T?

Notice that since the equation of the circle is x^2+y^2=1, then it has a radius of 1.

This means that OT = 1 (the length).
So if you draw out a right angle triangle, by drawing the perpendicular of T to the x-axis (lets call this point M), you have a right angle triangle with hypotenuse 1 and angle theta.
So

sin theta = MT / OT

Which gives the y-coordinate of T
Also

cos theta = MO / OT

So the x-coordinate is cos theta.

Alternatively:







(by trig properties)

And we can then find x, then y.

 

[Hypem]

Notice that since the equation of the circle is x^2+y^2=1, then it has a radius of 1.

This means that OT = 1 (the length).
So if you draw out a right angle triangle, by drawing the perpendicular of T to the x-axis (lets call this point M), you have a right angle triangle with hypotenuse 1 and angle theta.
So

sin theta = MT / OT

Which gives the y-coordinate of T
Also

cos theta = MO / OT

So the x-coordinate is cos theta.

Alternatively:







(by trig properties)

And we can then find x, then y.

Yeah I don't really get any of that lol, thanks for helping though!

I just realised that the x and y coordinate ARE the lengths and are interchangeable.
x-length (or coordinate) is cos and y-length (or coordinate) is sin, and since the radius is only 1 then it's only 1cos(theta) and 1sin(theta), so (cos(theta), sin(theta))

 

[Sy123]

Yeah I don't really get any of that lol, thanks for helping though!

I just realised that the x and y coordinate ARE the lengths and are interchangeable.
x-length (or coordinate) is cos and y-length (or coordinate) is sin, and since the radius is only 1 then it's only 1cos(theta) and 1sin(theta), so (cos(theta), sin(theta))

Yes that's pretty much what the first method was but the logic behind it.
But they are not interchangeable with respect to the theta that they have specified in the question. Parametrically speaking yes they are interchangeable but the thetas are different.

If the co-ordinates were (sin(theta), cos(theta)), then theta would be the angle OT makes with the y-axis.

 

[Hypem]

Not really sure what you mean? I meant they're interchangeable because we're measuring from the origin (0,0)

Like the x-coordinate and x-length are interchangeable, and the y-coordinate y-length are interchangeable. Obviously if you changed where the theta was positioned then how you'd find the x-length and y-length would change, but the coordinates would still be interchangeable with the lengths