How do you find the tangent lines of parametric curves?

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How do you find the tangent lines of parametric curves?

Show that the curve with parametric equations $x = sin t$ $x=sint$ , $y = sin (t + sin t)$ $y=sin(t+sint)$ has two tangent lines at the origin and find their equations. Illustrate by graphing the curve and its tangents.

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Answer 1 · 2018-01-16T17:50:51

$y = 2 x$ $y=2x$ and $y = 0$ $y=0$

Explanation:

$x = sin (t), y = sin (t + sin (t))$ $x=sin(t), y=sin(t+sin(t))$

First find the derivative:

$\frac{d x}{d t} = cos t$ $dx/dt=cost$

$\frac{d y}{d t} = (1 + cos t) cos (t + sin t)$ $dy/dt = (1+cost)cos(t+sint)$

$\frac{d y}{d x} = \frac{d y}{d t} \cdot \frac{d t}{d x} = \frac{(1 + cos t) cos (t + sin t)}{cos t}$ $dy/dx=dy/dt*dt/dx=((1+cost)cos(t+sint))/cost$

At the origin we have: $x = 0$ $x=0$ and $y = 0$ $y=0$ so from the x-coordinate:

$0 = sin (t) \to t_{0} = 0, t_{1} = π$ $0=sin(t)-> t_0=0, t_1=pi$

(There are of course more values which may satisfy this but the periodicity of the sine and cosine functions means that the other solutions such as $t=2pi,4pi,...$ will return the same tangent as $t=0$ and $t=3pi,5pi,...$ will return the same tangent as $t=pi$ so we need only consider the two solutions stated).

Checking that these values of $t$ satisfy $y=0$

$y=sin(0+sin(0))=0$
$y=sin(pi+sin(pi))=sin(pi+0)=0$

So $y$ is also satisfied. Put these values of $t$ into $dy/dx$ to find the gradient of the tangent line:

For $t=0$ :
$(dy/dx)_0=((1+cos(0))cos(0+sin0))/cos(0)=(2*cos(0))/cos(0)=2$

For $t=pi$

$(dy/dx)_1=((1+cos(pi))cos(pi+sinpi))/cos(pi)$

$(dy/dx)_1=((1-1)cos(pi))/cos(pi)=0$

So we have the gradients to our two tangent lines:

$m_(tan0)=2$ and $m_(tan1)=0$

We also have a point at which the lines pass through, the origin: $(0,0)$

Obviously, for the equation of a straight line $y=mx+c$ any line which passes through the origin will simply have $c=0.$ Therefore our two tangents will be:

$y_1=2x$ and $y=0$

When graphed, they look like this:

Generated on Mathematica

The blue line is the original parametric function and, the red line is $y=2x$ and the green line (lying right on the x-axis) is $y=0$ .

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How do you find the tangent lines of parametric curves?

Show that the curve with parametric equations $x = sin t$ $x=sint$ , $y = sin (t + sin t)$ $y=sin(t+sint)$ has two tangent lines at the origin and find their equations. Illustrate by graphing the curve and its tangents.

1 Answer

Explanation:

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How do you find the tangent lines of parametric curves?

Show that the curve with parametric equations x=sintx=sintx=sint, y=sin(t+sint)y=sin(t+sint)y=sin(t+sint) has two tangent lines at the origin and find their equations. Illustrate by graphing the curve and its tangents.

1 Answer

Explanation:

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Impact of this question

Show that the curve with parametric equations $x = sin t$ $x=sint$ , $y = sin (t + sin t)$ $y=sin(t+sint)$ has two tangent lines at the origin and find their equations. Illustrate by graphing the curve and its tangents.