Three quotients, three domains The reciprocal functions expand the circular family: $$\cot x=\frac{\cos x}{\sin x},\qquad \sec x=\frac1{\cos x},\qquad\csc x=\frac1{\sin x}.$$ Function Points excluded Image Fundamental period $\cot x$ $x=k\pi$ $\mathbb R$ $\pi$ $\sec x$ $x=\pi/2+k\pi$ $(-\infty,-1]\cup[1,\infty)$ $2\pi$ $\csc x$ $x=k\pi$ $(-\infty,-1]\cup[1,\infty)$ $2\pi$ In the entire table, $k\in\mathbb Z$. The cotangent is not defined by $1/\tan x$ in its entire domain: in $x=\pi/2$, the cotangent is worth 0, while the tangent does not exist. The identity $\cot x=1/\tan x$ requires that both expressions exist.
Compare coordinates In the first quadrant, the straight line of the radius that passes through $P=(\cos x,\sin x)$ has a slope equal to the height divided by the horizontal displacement. We extend this ratio to all angles where the denominator is not zero: $$\tan x=\frac{\sin x}{\cos x}.$$Logo: $$\operatorname{Dom}(\tan)=\mathbb R\setminus \left\{\frac\pi2+k\pi:k\in\mathbb Z\right\}.$$The image is $\mathbb R$. The tangent is positive in quadrants I and III and negative in quadrants II and IV. Its zeros are $x=k\pi$.
The width with sign For $P=(u,v)$ in the cycle, we define $\cos x=u$. The cosine is positive in the right semicircle and negative in the left. Its domain is $\mathbb R$ and its image is $[-1,1]$. The axis points provide: $x$ $0$ $\pi/2$ $\pi$ $3\pi/2$ $2\pi$ $\cos x$ $1$ $0$ $-1$ $0$ $1$ Zeros are $x=\pi/2+k\pi$. The maxima occur at $x=2k\pi$ and the minima at $x=\pi+2k\pi$, with $k\in\mathbb Z$. Demonstration of parity and period The points of $x$ and $-x$ are reflections on the horizontal axis. Its horizontal coordinate is equal:
The height of the point If the image of the real $x$ in the cycle is $P=(u,v)$, we define $\sin x=v$. The sine is the vertical, signed coordinate, not the unsigned distance to the horizontal axis. Any real has an image in the cycle. The height varies from $-1$ to 1, including the extremes. Therefore: $$\operatorname{Dom}(\sin)=\mathbb R,\qquad\operatorname{Im}(\sin)=[-1,1].$$In each turn, the sine is positive in the upper semicircle and negative in the lower one. Its zeros are $x=k\pi$, to $k\in\mathbb Z$.
From movement to a function A function associates a single output with each input of the domain. When making a point go through the trigonometric cycle, we can observe its height, its horizontal position or other geometrically constructed quantities. Each choice produces a circular function. The input will generally be a real number representing the spin in radians. Even if two different inputs lead to the same point, each has a unique image per function. A function can repeat outputs.
Compare an arc with its radius If an arc has length $s$ in a circle of radius $R>0$, its measure in radians is: $$\theta=\frac{s}{R}.$$A radian is the central angle that cuts an arc as long as the radius. Since the length of a circle is $2\pi R$, one revolution measures $2\pi$ radians. It follows that: $$360°=2\pi\text{ rad},\qquad180°=\pi\text{ rad}.$$The conversions are $\theta_{\mathrm{rad}}=\theta_{\mathrm{graus}}\pi/180$ and $\theta_{\mathrm{graus}}=\theta_{\mathrm{rad}}180/\pi$. For example, $75°=5\pi/12$ rad. Demonstrate the length formula An arc of $\alpha$ degrees occupies fraction $\alpha/360$ of the turn. Its length is: