1
$\begingroup$

I am just starting to learn mathematics though self study and I am curious about the following:

Define $S=\{(x_1,x_2)\in\mathbb{R^2}:x_1^2+x_2^2=1\}$. How can I show (directly from the definition) that the map $f:\mathbb{R}\to S$ where $f(t)=\big(\cos(2\pi t), \sin(2\pi t)\big)$ is surjective?

Thank you in advance.

1 Answers 1

1

Since:

1) $x_1^2+x_2^2=1\,\Longrightarrow \,|x_1|\,,\,|x_2|\leq 1\,\Longleftrightarrow -1\leq x_1\,,\,x_2\leq 1$ ;

2) Both functions $\,\sin x\,,\,\cos x\,$ are onto $\,[-1,1]\,$ ;

3) By The Trigonometric Pythagoras Theorem, $\,\sin^2 x+\cos^2 x=1\,$ ,

then...