- #1
cmut
- 2
- 0
I have tirelessly tried to solve this out seems i need smnes help: if x=2+7cosθ and y=8+3sinθ show that d2y/dx2=(-3cosec3θ)/49
Solve Parametric Equations | Step-by-Step Guide
- Thread starter cmut
- Start date
In summary, the given equations x=2+7cosθ and y=8+3sinθ can be used to derive d2y/dx2=(-3cosec3θ)/49 by using the quotient rule and differentiating with respect to θ. The final result is -(3/49)csc^3(θ).
- #2
mfb
Mentor
- 37,220
- 14,067
You could post what you tried so far.
You can derive an expression y=f(x). Alternatively, there is a nice way to get an expression f(x,y)=0, which can be derived afterwards.
- #3
cmut
- 2
- 0
I used the formula for d2y/dx2=(d/dθ dy/dx)/dx/dθ, i further used the quotient rule to simplify the expression and found they are not the same i got 21cosec2θ/49...it seems right but would you think otherwise? i greatly appreciate your feedback Mfb
- #4
mfb
Mentor
- 37,220
- 14,067
Okay, I checked it myself, and I get the same result as you.
- #5
HallsofIvy
Science Advisor
Homework Helper
- 42,988
- 975
dx/dθ= -7 sin(θ) and dy/dθ= 3 cos(θ) so dy/dx= (-3/7) cot(θ)cmut said:
Then d^2y/dx= d/dx(-(3/7) cot(θ))= (3/7) csc^2(θ) dθ/dx= (3/7) csc^2(θ)/(-7 sin(θ))= -(3/49)csc^3(θ)
FAQ: Solve Parametric Equations | Step-by-Step Guide
What are parametric equations?
Parametric equations are a set of equations that express a set of coordinates (x, y) in terms of one or more parameters, typically denoted by t. This allows for a more versatile way of representing curves and allows for the manipulation of the curve by changing the parameters.
Why do we use parametric equations?
Parametric equations are used to represent curves that cannot be expressed in the form of a function, such as circles, ellipses, and spirals. They also allow for more control and manipulation of the curve by changing the parameters, making them useful in many applications in physics, engineering, and science.
How do you solve parametric equations?
To solve parametric equations, you can use the same techniques as you would for solving regular equations. The first step is to eliminate the parameter (usually t) by solving for it in one equation and substituting it into the other equation. This will give you an equation in terms of x and y, which you can then solve using algebraic techniques.
What are some common mistakes when solving parametric equations?
One common mistake is forgetting to eliminate the parameter and trying to solve for x and y independently. Another mistake is not properly simplifying the equations before solving, which can lead to incorrect solutions. It's also important to pay attention to the domain of the parameter and make sure to account for any restrictions when solving the equations.
How can I check if my solution to a parametric equation is correct?
You can check your solution by graphing the parametric equations and seeing if the points you obtained from solving the equations lie on the graph. You can also substitute the values of x and y back into the original parametric equations to see if they satisfy them. It's always a good idea to double-check your work to ensure accuracy.