This sketch shows how we can create a function for tangent.[br][br]As a point goes around the unit circle, we can see how the different ratios change based on the angle, given here as [math]x[/math] and measured in radians.[br][br]First, select Tangent and then Animate. At each of the points on the traced out curve, the [math]x[/math]-coordinate represents the angle and the [math]y[/math]-coordinate represents the sine of the angle.[br][br]The tangent of the angle can be found on using the point on the unit circle: [br][math]\tan\theta=\frac{y}{x}[/math][br]In other words, the tangent gives us the slope of the terminal side of the angle.[br][br]Also, if we were to draw two lines tangent to our unit circle at [math](1,0)[/math] and [math](-1,0)[/math], then the leg of the right triangle formed with the radius from [math]\left(0,0\right)[/math] to [math]\left(1,0\right)[/math] and the terminal side of [math]x[/math] would have a length of [math]\tan x[/math].