# Drawing with trigonometric function

## Homework Statement

Come up with a set of equations to draw the following picture:

http://img197.imageshack.us/img197/7428/equation.jpg [Broken]

## The Attempt at a Solution

The circle is easy.

1000 = sqrt(x^2+y^2)

I can just model the equation with sines. So

y_1 = 300 * Sin(pi * x / 1000) (Assuming 300 is the highest point in the curve)

The sine graph on the y axis can be easily tilted by switching x and y. Therefore:

x = 300 * Sin(pi*y_2 / 1000)

sin^-1 (x/300) = pi * y_2 / 1000

y_2 = (1000/pi) * sin^-1(x/300)

Is my y_2 correct?

And I'm kinda stumbling on how to translate the axis in 45 degree angle to draw the sine graph like that. Any help would be appreciated.

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eumyang
Homework Helper
I can just model the equation with sines. So

y_1 = 300 * Sin(pi * x / 1000) (Assuming 300 is the highest point in the curve)
You should probably add a restriction for x, like -1000 ≤ x ≤ 1000.

The sine graph on the y axis can be easily tilted by switching x and y. Therefore:

x = 300 * Sin(pi*y_2 / 1000)

sin^-1 (x/300) = pi * y_2 / 1000

y_2 = (1000/pi) * sin^-1(x/300)

Is my y_2 correct?
Technically, no, because x has the domain restriction -π/2 ≤ x ≤ π/2. Must all equations be y as functions of x? If not, I would just leave it in terms of x. Also, I think you need a negative in front of the R.H.S.:
x = -300 * Sin(pi*y_2 / 1000)

No it can be in terms of X too.

How should I modelize the lines which goes at 45 degree angle?

eumyang
Homework Helper
The circle is easy.

1000 = sqrt(x^2+y^2)

No it can be in terms of X too.

How should I modelize the lines which goes at 45 degree angle?
Are you familiar with rotation of axes? You'll need to make use of it, I think.

Mentallic
Homework Helper
No it's not,

The circle is easy.

1000 = sqrt(x^2+y^2)

:tongue:

Are you familiar with rotation of axes? You'll need to make use of it, I think.

Yeah, translating the axes by defining a new set of axes such as x' and y', then rotating it via trigonometric function but I was struggling big time about that. Any suggestions?

eumyang
Homework Helper
No it's not,
:tongue:
Wow... that's was embarrassing. Let's try that again.
The circle is easy.

1000 = sqrt(x^2+y^2)
This only gives you the upper semicircle. Define this implicitly instead:
x2+y2=10002

Yeah, translating the axes by defining a new set of axes such as x' and y', then rotating it via trigonometric function but I was struggling big time about that. Any suggestions?
Do you know the equations that define x' and y' in terms of x, y, and θ? Take the equation
y' = 300 * sin(pi*x'/1000)
and replace with what x' and y' equals. Then find a new set of equations for x' and y', in terms of a different angle θ, and replace into the equation above.

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LCKurtz
$$R(x) = \left[ \begin{array}{cc} \cos\theta & -\sin\theta\\ \sin\theta & \cos\theta \end{array}\right] \left[ \begin{array}{c} x\\ f(x) \end{array}\right]$$