# Why does tan(t) touch the unit circle at (1,0)?

I can't get my head around this, any help would be very much appreciated. Thanks

EDIT: t is an angle, where 0 < t < 90, angle t is in degrees EDIT: Added a picture I lifted from google

• Please clarify: What is $t$? – Ryuky Mar 25 '13 at 12:48
• an angle, 0<t<90 – seeker Mar 25 '13 at 12:51
• angle in radians or degrees? – Arjang Mar 25 '13 at 12:53
• It's in degrees – seeker Mar 25 '13 at 13:00
• For convenience. You could well define another function, puting this line anywhere, but it wouldn't give all these pretty formulas. By the way, if your line touches $(0,1)$, then you get cotangent. – Jean-Claude Arbaut Mar 25 '13 at 13:52

I think what you're trying to express is that the length of the vertical segment joining $P=(1,0)$ to the point labelled $R$ in the diagram is the value of $\tan h$ where $h$ is the angle in the diagram. When $R$ is above the $x$ axis (as in the diagram), this is true by using the definition of $\tan h$ as "opposite over adjacent", because in this picture the "adjacent" has length $1$, being the segment from $O=(0,0)$ to $P=(1,0).$

The same thing works if $R$ happens to be below the $x$ axis, provided we interpret the segment joining $P$ to $R$ as having negative length (i.e. "signed length" is negative) because it extends downward from $P$ Think of the vertical line through $P$ as a real axis, labelled with numbers going positive in the upward direction.

• That seems to have jolted some neurons! I get it, thanks! – seeker Mar 25 '13 at 13:36

What the picture shows is that the right triangle $OSQ$ is similar to the right triangle $OPR$. In particular, $$\tan(h)=\frac{\sin(h)}{\cos(h)}$$ by definition, and trivially, $$\tan(h)=\frac{\tan(h)}1.$$ The length of $OS$ is $\cos(h)$ ahd the length of $SQ$ is $\sin(h)$ by definition. Since it's a unit circle, then the length of $OP$ is $1$, so by similarity, the length of $PR$ is therefore $\tan(h)$.

• Exactly! The $\tan(t)$ that lies tangent to the circle isn't the function of the graph. It's just the length of the line. – Brian Silva Mar 25 '13 at 13:36

Let us find the intersection of the Unit Circle is $x^2+y^2=1$ with the line $y=\tan t$

So, $x^2=1-y^2=1-\tan^2t\implies x^2+\tan^2t-1=0$

This is a Quadratic equation in $x$

For tangency, the two roots must be equal i.e., the discriminant $(0)^2-4\cdot1\cdot(\tan^2t-1)$ must be $0$

$\implies \tan^2t=1\implies \tan t=1$ as $0^\circ<t<90^\circ$

$\implies x=0, y=1$

So, $y=\tan t$ is tangent of the Unit Circle at $(0,1)$