# How do you find the tangent lines of parametric curves?

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

Show that the curve with parametric equations

##### 1 Answer

#### Explanation:

First find the derivative:

At the origin we have:

(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

Checking that these values of

So

For

For

So we have the gradients to our two tangent lines:

We also have a point at which the lines pass through, the origin:

Obviously, for the equation of a straight line

When graphed, they look like this:

The blue line is the original parametric function and, the red line is