# Tables of trigonometric functions

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## Properties of trigonometric functions

### Representation by other trigonometric functions

${\mathrm{sin}}^{2}x$

${\mathrm{cos}}^{2}x$

${\mathrm{tan}}^{2}x$

${\mathrm{cot}}^{2}x$

${\mathrm{sin}}^{2}x$

-

$1-{\mathrm{cos}}^{2}x$

$\frac{{\mathrm{tan}}^{2}x}{1+{\mathrm{tan}}^{2}x}$

$\frac{1}{1+{\mathrm{cot}}^{2}x}$

${\mathrm{cos}}^{2}x$

$1-{\mathrm{sin}}^{2}x$

-

$\frac{1}{1+{\mathrm{tan}}^{2}x}$

$\frac{{\mathrm{cot}}^{2}x}{1+{\mathrm{cot}}^{2}x}$

${\mathrm{tan}}^{2}x$

$\frac{{\mathrm{sin}}^{2}x}{1-{\mathrm{sin}}^{2}x}$

$\frac{1-{\mathrm{cos}}^{2}x}{{\mathrm{cos}}^{2}x}$

-

$\frac{1}{{\mathrm{cot}}^{2}x}$

${\mathrm{cot}}^{2}x$

$\frac{1-{\mathrm{sin}}^{2}x}{{\mathrm{sin}}^{2}x}$

$\frac{{\mathrm{cos}}^{2}x}{1-{\mathrm{cos}}^{2}x}$

$\frac{1}{{\mathrm{tan}}^{2}x}$

-

### Function values of trigonometric functions

0

$\frac{\pi }{6}$

$\frac{\pi }{4}$

$\frac{\pi }{3}$

$\frac{\pi }{2}$

Degree

30°

45°

60°

90°

${\mathrm{sin}}^{2}x$

0

$\frac{1}{2}$

$\frac{1}{2}\sqrt{2}$

$\frac{1}{2}\sqrt{3}$

1

${\mathrm{cos}}^{2}x$

1

$\frac{1}{2}\sqrt{3}$

$\frac{1}{2}\sqrt{2}$

$\frac{1}{2}$

0

${\mathrm{tan}}^{2}x$

0

$\frac{1}{3}\sqrt{3}$

1

$\sqrt{3}$

-

${\mathrm{cot}}^{2}x$

-

$\sqrt{3}$

1

$\frac{1}{3}\sqrt{3}$

0

### Reduction formulas (in degrees)

$\mathrm{sin}\left(90°+x\right)$$=\mathrm{cos}\left(x\right)$

$\mathrm{cos}\left(90°+x\right)$$=-\mathrm{sin}\left(x\right)$

$\mathrm{tan}\left(90°+x\right)$$=-\mathrm{cot}\left(x\right)$

$\mathrm{cot}\left(90°+x\right)$$=-\mathrm{tan}\left(x\right)$

$\mathrm{sin}\left(180°+x\right)$$=-\mathrm{sin}\left(x\right)$

$\mathrm{cos}\left(180°+x\right)$$=-\mathrm{cos}\left(x\right)$

$\mathrm{tan}\left(180°+x\right)$$=\mathrm{tan}\left(x\right)$

$\mathrm{cot}\left(180°+x\right)$$=\mathrm{cot}\left(x\right)$

$\mathrm{sin}\left(\frac{\pi }{2}+x\right)$$=\mathrm{cos}\left(x\right)$

$\mathrm{cos}\left(\frac{\pi }{2}+x\right)$$=-\mathrm{sin}\left(x\right)$

$\mathrm{tan}\left(\frac{\pi }{2}+x\right)$$=-\mathrm{cot}\left(x\right)$

$\mathrm{cot}\left(\frac{\pi }{2}+x\right)$$=-\mathrm{tan}\left(x\right)$

$\mathrm{sin}\left(\pi +x\right)$$=-\mathrm{sin}\left(x\right)$

$\mathrm{cos}\left(\pi +x\right)$$=-\mathrm{cos}\left(x\right)$

$\mathrm{tan}\left(\pi +x\right)$$=\mathrm{tan}\left(x\right)$

$\mathrm{cot}\left(\pi +x\right)$$=\mathrm{cot}\left(x\right)$

### Context of the trigonometric functions for the same argument

${\mathrm{sin}\left(x\right)}^{2}+{\mathrm{cos}\left(x\right)}^{2}$$=1$

$\mathrm{tan}\left(x\right)$$=\frac{\mathrm{sin}\left(x\right)}{\mathrm{cos}\left(x\right)}$

$\mathrm{cot}\left(x\right)$$=\frac{\mathrm{cos}\left(x\right)}{\mathrm{sin}\left(x\right)}$$=\frac{1}{\mathrm{tan}\left(x\right)}$

$\mathrm{cot}\left(x\right)\cdot \mathrm{tan}\left(x\right)$$=1$

### Addition theorems of trigonometric functions

$\mathrm{sin}\left(x±y\right)$$=\mathrm{sin}\left(x\right)\mathrm{cos}\left(y\right)±\mathrm{cos}\left(x\right)\mathrm{sin}\left(y\right)$

$\mathrm{cos}\left(x±y\right)$$=\mathrm{cos}\left(x\right)\mathrm{cos}\left(y\right)\mp \mathrm{sin}\left(x\right)\mathrm{sin}\left(y\right)$

$\mathrm{tan}\left(x±y\right)$$=\frac{\mathrm{tan}\left(x\right)±\mathrm{tan}\left(y\right)}{1\mp \mathrm{tan}\left(x\right)\mathrm{tan}\left(y\right)}$

### Addition theorems for multiples of the argument value

$\mathrm{sin}\left(2x\right)$$=2\mathrm{sin}\left(x\right)\mathrm{cos}\left(x\right)$

$\mathrm{sin}\left(3x\right)$$=3\mathrm{sin}\left(x\right)-4{\mathrm{sin}}^{3}\left(x\right)$

$\mathrm{cos}\left(2x\right)$$={\mathrm{cos}}^{2}\left(x\right)-{\mathrm{sin}}^{2}\left(x\right)$

$\mathrm{cos}\left(3x\right)$$=4{\mathrm{cos}}^{3}\left(x\right)-3\mathrm{cos}\left(x\right)$

### Half of the argument value

$\mathrm{sin}\left(\frac{x}{2}\right)$$=±\sqrt{\frac{1-\mathrm{cos}\left(x\right)}{2}}$

$\mathrm{cos}\left(\frac{x}{2}\right)$$=±\sqrt{\frac{1+\mathrm{cos}\left(x\right)}{2}}$

$\mathrm{tan}\left(\frac{x}{2}\right)$$=±\sqrt{\frac{1-\mathrm{cos}\left(x\right)}{1+\mathrm{cos}\left(x\right)}}$

### Sum and difference of trigonometric functions

$\mathrm{sin}\left(x\right)+\mathrm{sin}\left(y\right)$$=2\mathrm{sin}\left(\frac{x+y}{2}\right)\mathrm{cos}\left(\frac{x-y}{2}\right)$

$\mathrm{cos}\left(x\right)+\mathrm{cos}\left(y\right)$$=2\mathrm{cos}\left(\frac{x+y}{2}\right)\mathrm{cos}\left(\frac{x-y}{2}\right)$

$\mathrm{sin}\left(x\right)-\mathrm{sin}\left(y\right)$$=2\mathrm{cos}\left(\frac{x+y}{2}\right)\mathrm{sin}\left(\frac{x-y}{2}\right)$

$\mathrm{cos}\left(x\right)-\mathrm{cos}\left(y\right)$$=-2\mathrm{sin}\left(\frac{x+y}{2}\right)\mathrm{sin}\left(\frac{x-y}{2}\right)$

$\mathrm{cos}\left(x\right)±\mathrm{sin}\left(x\right)$$=\sqrt{2}\mathrm{sin}\left(\frac{\pi }{4}±x\right)$$=\sqrt{2}\mathrm{cos}\left(\frac{\pi }{4}\mp x\right)$

$\mathrm{tan}\left(x\right)±\mathrm{tan}\left(y\right)$$=\frac{\mathrm{sin}\left(x±y\right)}{\mathrm{cos}\left(x\right)\mathrm{cos}\left(y\right)}$

$\mathrm{cot}\left(x\right)±\mathrm{cot}\left(y\right)$$=±\frac{\mathrm{sin}\left(x±y\right)}{\mathrm{sin}\left(x\right)\mathrm{sin}\left(y\right)}$

$\mathrm{tan}\left(x\right)+\mathrm{cot}\left(y\right)$$=\frac{\mathrm{cos}\left(x-y\right)}{\mathrm{cos}\left(x\right)\mathrm{sin}\left(y\right)}$

$\mathrm{cot}\left(x\right)-\mathrm{tan}\left(y\right)$$=\frac{\mathrm{cos}\left(x+y\right)}{\mathrm{sin}\left(x\right)\mathrm{cos}\left(y\right)}$

### Products of trigonometric functions

$\mathrm{sin}\left(x\right)\mathrm{sin}\left(y\right)$$=\frac{1}{2}\left(\mathrm{cos}\left(x-y\right)-\mathrm{cos}\left(x+y\right)\right)$

$\mathrm{cos}\left(x\right)\mathrm{cos}\left(y\right)$$=\frac{1}{2}\left(\mathrm{cos}\left(x-y\right)+\mathrm{cos}\left(x+y\right)\right)$

$\mathrm{sin}\left(x\right)\mathrm{cos}\left(y\right)$$=\frac{1}{2}\left(\mathrm{sin}\left(x-y\right)+\mathrm{sin}\left(x+y\right)\right)$

$\mathrm{cos}\left(x\right)\mathrm{sin}\left(y\right)$$=\frac{1}{2}\left(\mathrm{sin}\left(x+y\right)-\mathrm{sin}\left(x-y\right)\right)$

### Potencies of trigonometric functions

${\mathrm{sin}}^{2}\left(x\right)$$=\frac{1}{2}\left(1-\mathrm{cos}\left(2x\right)\right)$

${\mathrm{sin}}^{3}\left(x\right)$$=\frac{1}{4}\left(3\mathrm{sin}\left(x\right)-\mathrm{sin}\left(3x\right)\right)$

${\mathrm{sin}}^{n}\left(x\right)$$=\frac{1}{{2}^{n}}\sum _{k=0}^{n}\left(\begin{array}{c}n\\ k\end{array}\right)\mathrm{cos}\left(\left(n-2k\right)\left(x-\frac{\pi }{2}\right)\right)$

${\mathrm{cos}}^{2}\left(x\right)$$=\frac{1}{2}\left(1+\mathrm{cos}\left(2x\right)\right)$

${\mathrm{cos}}^{3}\left(x\right)$$=\frac{1}{4}\left(3\mathrm{cos}\left(x\right)+\mathrm{cos}\left(3x\right)\right)$

${\mathrm{cos}}^{n}\left(x\right)$$=\frac{1}{{2}^{n}}\sum _{k=0}^{n}\left(\begin{array}{c}n\\ k\end{array}\right)\mathrm{cos}\left(\left(n-2k\right)x\right)$