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20/10/2021
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Basic definition of the trigonometric functions
Cosine function

Particular values of the cosine function (cos):

$$\displaystyle cos(0^{\circ})=cos(0)=1$$

$$\displaystyle cos(30^{\circ})=cos(\frac{\pi}{6})=\frac{\sqrt{3}}{2}$$

$$\displaystyle cos(45^{\circ})=cos(\frac{\pi}{4})=\frac{\sqrt{2}}{2}$$

$$\displaystyle cos(60^{\circ})=cos(\frac{\pi}{3})=\frac{1}{2}$$

$$\displaystyle cos(90^{\circ})=cos(\frac{\pi}{2})=0$$

$$\displaystyle cos(x + 2\pi)=cos(x)$$

Sine function

Particular values of the sine function (sin):

$$\displaystyle sin(0^{\circ})=sin(0)=0$$

$$\displaystyle sin(30^{\circ})=sin(\frac{\pi}{6})=\frac{1}{2}$$

$$\displaystyle sin(45^{\circ})=sin(\frac{\pi}{4})=\frac{\sqrt{2}}{2}$$

$$\displaystyle sin(60^{\circ})=sin(\frac{\pi}{3})=\frac{\sqrt{3}}{2}$$

$$\displaystyle sin(90^{\circ})=sin(\frac{\pi}{2})=1$$

$$\displaystyle sin(x + 2\pi)=sin(x)$$

Tangent function

Particular values of the tangent function (tan):

$$\displaystyle tan(0^{\circ})=tan(0)=0$$

$$\displaystyle tan(30^{\circ})=tan(\frac{\pi}{6})=\frac{\sqrt{3}}{3}$$

$$\displaystyle tan(45^{\circ})=tan(\frac{\pi}{4})=1$$

$$\displaystyle tan(60^{\circ})=tan(\frac{\pi}{3})=\sqrt{3}$$

The tangent function is not defined at $$\displaystyle \frac{\pi}{2}$$ ($$\displaystyle 90^{\circ}$$).

$$\displaystyle tan(x + \pi)=tan(x)$$

Relationships between trigonometric functions

Main relationships between trigonometric functions:

$$\displaystyle cos^2(x)+sin^2(x)=1$$

$$\displaystyle tan(x)=\frac{sin(x)}{cos(x)}$$

$$\displaystyle \frac{1}{cos^2(x)}=1+tan^2(x)$$

$$\displaystyle cotan(x)=\frac{1}{tan(x)}$$

$$\displaystyle cotan(x)=\frac{cos(x)}{sin(x)}$$

$$\displaystyle \frac{1}{sin^2(x)}=1+cotan^2(x)$$