Question

Find an equation of the tangent to the curve at the point x= cos t + cos (2t) , y = sin t + sin (2t), (-1,1)

Answer 1

\[\frac{ dy }{ dx } = \frac{ dy / dt }{ dx/dt }\]
\[\frac{ dy }{ dt} = \cos t + 2 \cos(2t)\]
\[\frac{ dx }{ dt} = -\sin t -2\sin(2t)\]
\[\frac{ dy }{ dx } = \frac{ \cos t + 2 \cos (2t) }{ -\sin t -2\sin(2t) }\]

Answer 2

Kind of confused on what to do right now. How to simplify all this trig. D:

Answer 3

dont worry about the simplification, we simply need to determine that t value

Answer 4

assuming -1,1 is the stated point

what is the value of t, when does x=-1, and y=1?

Answer 5

-1= cos t + cos (2t)
1 = sin t + sin (2t)

adding them might prove beneficial

0 = cos(t) + sin(t) + cos(2t) + sin(2t)

Answer 6

I didn't even think that you could do that. Wow...

Answer 7

Is that a double angle identity?

Answer 8

no, its just x+y, given that x=-1 and y=1

Answer 9

we could just work one part of it ... all depends on how simple we can make it for ourselves

1 = cos(t) + cos(2t)

what are our solutions for t?

Answer 10

er, -1 that is

Answer 11

do you understand what we are doing at this point? if we know t, we can determine the value of our derivatives.

Answer 12

No, no I don't. Have am I in Calc. 3 and not know this...

Answer 13

to form a line, we need a slope and a point, slope = dy/dx

\[\frac{ dy }{ dx } = \frac{ \cos t + 2 \cos (2t) }{ -\sin t -2\sin(2t) }\]
no need to work this out to death since all we really need is a value of t, and not some simplified rewritten formula for dy/dx

the point given is (x=-1, y=1)

so we simply need to assess the value of t for that point from our equations of x and y

Answer 14

Here is a graph of the problem