3. (a) Why does the curve with parametric equations x = sin...

3. (a) Why does the curve with parametric equations $x = \sin t$, $y = \sin(t + \sin t)$ have two tangent lines at the origin? (b) Find the equations of both tangent lines. (c) Use technology to help you sketch the graph of the curve and its two tangent lines at the origin.

Transcript

00:01 In this question, we are talking about the chain rule and answering this multiple choice question about the tangent to the curve given by these parametric equations.

00:13 That is the tangent at t equals zero.

00:19 We want to know which of these options is the equation of this tangent line.

00:27 So to think about this question, recall that the tangent line needs to pass through the same point.

00:39 Needs to pass through the point of tangency, obviously if it's going to be tangent, it must pass through the point at which it touches the curve.

00:52 And it must also have a slope that is the derivative of the curve at the point.

01:00 And in this case, since we're talking about a slope in the x, y plane, that derivative is the derivative of y with respect to x.

01:11 Is just a parameter, so it's not with respect to t.

01:21 What is this point of tangency that we're talking about? well, since we're considering t equals zero, and the x -coordinate at this point is going to be 0 minus cosine of 0, and the y -coordinate is going to be minus 1 plus sine of 0.

01:47 And so we might already try to find out which equation this is going to be.

01:56 We can already rule some out by considering whether or not they pass through the point negative 1, negative 1.

02:05 So let's start from the bottom and work our way up.

02:10 If x is negative 1 here is y equal to negative 1? no.

02:17 Oh, yes, actually.

02:19 So it might be part e, or option e.

02:25 What about d? here if we substitute x is negative 1, we don't get y as negative 1.

02:32 So it can't be d...

Question

3. (a) Why does the curve with parametric equations $x = \sin t$, $y = \sin(t + \sin t)$ have two tangent lines at the origin? (b) Find the equations of both tangent lines. (c) Use technology to help you sketch the graph of the curve and its two tangent lines at the origin.

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