In the context of the Unit Circle why is tan$(\theta)$ defined as $\tan(\theta)=\frac{\sin (\theta)}{\cos (\theta)}=\frac{y}{x}$?

Question

I understand why the circular functions $\sin(\theta)=y$ and $\cos(\theta)=x$, but why does $\tan(\theta)=\frac{\sin (\theta)}{\cos (\theta)}=\frac{y}{x}$? Is there any particular reason why $\tan (\theta)$ is defined as the ratio of $\sin (\theta)$ and $\cos (\theta)$? Furthermore why does the tangent line specifically touch the Unit Circle at the point $(1,0)$ (presumably this leads on from how $\tan(\theta)$ is defined)?

Answer 1

Edit: Personally I prefer all the other answers, here's another way to think about it if you want more ideas

If $x^2 + y^2 = 1$, then $2x + 2y \frac{dy}{dx} = 0$

Re-arranging, $\frac{dy}{dx} = \frac{-x}{y}$. As you say, $\cos \theta = x$ and $\sin \theta = y$, giving you the slope of a tangent line as $- \cot \theta$. You can then use the equation: $$y - \sin \theta = - \cot \theta (x - \cos \theta)$$

Answer 2

The tangent is orthogonal to the line which goes from the center to the point on which you want the tangent. As this point is $$(\cos(x),\sin(x))$$ the tangent. will be $$(-\sin(x),\cos(x))\cdot t+(\cos(x),\sin(x))$$

You can easily visualize this using an CAS like Mathematica, taking as input

Manipulate[
 ParametricPlot[
 {
  {Cos[Pi*t],Sin[Pi*t]}, 
  {Cos[x], Sin[x]} + {-Sin[x], Cos[x]}*t
 },
{t, -1, 1}], 
{x, -Pi, Pi}]

gives you a slider where you can go through every angle. If you like I can make some pictures.

A picture

What i am essentially doing is drawing at first the circle, than i take a point on the circle which is $(\cos(x),\sin(x))$. From here I make a line in the direction of $(-\sin(x),\cos(x))$. This essentialy is a line of the form $a t + b$ but here $a$ and $b$ are vectors.

Answer 3

Take a point $\,(\cos x_0\,,\,\sin x_0)\,$ on the unit circle, and assume $\,\cos x_0\cdot\sin x_0\neq 0\,$ (otherwise the question is almost trivial) , so the slope of the radius in the circle to this point is

$$\frac{\sin x_0}{\cos x_0}$$

Since the tangent line to the circle in the above point is perpendicular to the radius at that point, the tangent line's slope must have slope

$$-\frac{\cos x_0}{\sin x_0}$$

So now you have a point on the tangent line and its slope: calculate its formula.

Answer 4

I copied this image from http://www.sagemath.org. It helped me understand what all of the trig ratios represent.

Answer 5

From the article $148,150$ of The elements of coordinate geometry, by Loney, the equation of the tangent of the circle $x^2+y^2=a^2$ at $(x_1,y_1)$ is
$$xx_1+yy_1=a^2$$

As you have already identified, any point on the circle can be $(a\cos\theta,a\sin\theta)$ where $0\le \theta<2\pi$

So, the equation of the tangent becomes $$xa\cos\theta+ya\sin\theta=a^2\implies x\cos\theta+y\sin\theta=a\text{ as }a\ne0 $$

For the Unit Circle $a=1,$ So, the equation of the tangent becomes $x\cos\theta+y\sin\theta=1$