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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:

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

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

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

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

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

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

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