Question

I used a graphing tool to graph this for my assignment but I dont understand how I would solve it by hand x = 7 sin t + sin 7t and y = 7 cos t + cos 7t

Answer 1

What's there to solve

Answer 2

I mean you know how with say y=3sint and x=3cost you can use the trig proof of cos^2+sin^2=1 to find that this is the graph of a circle, but how would I do that with something like this

Answer 3

because if I divide by 7 I end up with the other sin/cos with 7 under it so I a having trouble figuring out how to isolate the t or find a trig ident to help me get the grpah which I know looks like say an eight leaf clover

Answer 4

or is this one of those equations where you need a calculator to solve it?

Answer 5

Here was my initial idea, but (DISCLAIMER) you'll find that it doesn't work out in the end.
\[\begin{cases}x=7\sin t+\sin7t\\
y=7\cos t+\cos7t\end{cases}~\Rightarrow~\begin{cases}x^2=49\sin^2t+14\sin t\sin7t+\sin^27t\\y^2=49\cos^2t+14\cos t\cos7t+\cos^27t\end{cases}\]
Then you have
\[x^2+y^2=49\left(\sin^2t+\cos^2t\right)+14(\sin t\sin7t+\cos t\cos7t)+\left(\sin^27t+\cos^27t\right)\]
which reduces to
\[x^2+y^2=49+14(\sin t\sin7t+\cos t\cos7t)+1\\
x^2+y^2=50+14(\sin t\sin7t+\cos t\cos7t)\]
Then use another identity:
\[\cos(\alpha\pm\beta)=\cos\alpha\cos\beta\mp\sin\alpha\sin\beta\]
which you can use to write the remaining trig term as
\[\cos t\cos7t+\sin t\sin7t=\cos(7-1)t=\cos6t\]
So you have
\[x^2+y^2=50+14\cos6t\]
I don't see any other way to graph this without a calculator.

Answer 6

that looks like that is it, but I have never heard of cos(α±β)=cosαcosβ∓sinαsinβ so maybe that is from a higher level math and for they do only want e to use a calculator but I really appreciate this thank you

Answer 7

I think the name is "angle sum/difference" identity or something like that. Like I said, what I wrote up there doesn't work out nicely (unless there's some way to remove \(t\) altogether).