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Analytic geometry

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Trigonometry

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$1^\prime=\dfrac{\pi}{180 \cdot 3600}\,\text{rad}\approx 0.000291\text{rad}$

Radian

Degree

$1^{\prime\prime}=\dfrac{\pi}{180 \cdot 3600}\,\text{rad}\approx0.000005\,\text{red}$

Degree

Radian

$\sin^2\alpha+\cos^2\alpha=1$

Trigonometry

$\tan\alpha=\dfrac{\sin\alpha}{\cos\alpha}$

Trigonometry

$\cot\alpha=\dfrac{\cos\alpha}{\sin\alpha}$

Trigonometry

$\tan\alpha\cdot\cot\alpha=1$

Trigonometry

$\tan^2\alpha=\dfrac{1}{\cos^2\alpha}-1$

Trigonometry

Sine of sum of angles

$\sin(\alpha+\beta)=\sin\alpha \, \cos\beta + \sin\beta \, \cos\alpha$

Trigonometry

Example

If $\alpha = 30\degree$ and $\beta= 45\degree$ $$\sin(30\degree+45\degree)=\sin30\degree \, \cos45\degree + \sin45\degree \, \cos30\degree =\\= \dfrac{1}{2}\cdot\dfrac{\sqrt{2}}{2}+\dfrac{\sqrt{2}}{2}\cdot \dfrac{\sqrt{3}}{2}=\dfrac{\sqrt{2}+\sqrt{6}}{4}$$

$\sin(\alpha - \beta) = \sin\alpha \, \cos\beta - \sin\beta\,\cos\alpha$

Trigonometry

10.

$\cos(\alpha+\beta)=\cos\alpha\cos\beta-\sin\alpha\sin\beta$

Trigonometry

11.

$\cos(\alpha-\beta)=\cos\alpha\cos\beta+\sin\alpha\sin\beta$

Trigonometry

12.

$\tan(\alpha+\beta)=\dfrac{\tan\alpha + \tan\beta}{1-\tan\alpha\tan\beta}$

Trigonometry

13.

$\tan(\alpha-\beta)=\dfrac{\tan\alpha - \tan\beta}{1+\tan\alpha\tan\beta}$

Trigonometry

14.

$\cot(\alpha+\beta)=\dfrac{1-\tan\alpha\tan\beta}{\tan\alpha+\tan\beta}$

Trigonometry

15.

$\cot(\alpha-\beta)=\dfrac{1+\tan\alpha\tan\beta}{\tan\alpha-\tan\beta}$

Trigonometry

16.

Difference of fifth powers

$a^5-b^5=(a-b)(a^4+a^3b+a^2b^2+ab^3+b^4)$

Factoring

17.

Sum of fifth powers

$a^5+b^5=(a+b)(a^4-a^3b+a^2b^2-ab^3+b^4)$

Factoring

Polynomial

Description

This formula shows how to factor the sum of two fifth powers.

18.

Square of difference

$(a-b)^2=a^2-2ab+b^2$

Polynomial

Power

Example

Example 1:| $(a-2)^2 = a^2 - 2\cdot a \cdot 2 + 2^2 = a^2 - 4a + 4$ Example 2:| $A (3x-5y)^2 &= (3x)^2 - 2 \cdot 3x \cdot 5y + (5y)^2\\ &= 9x^2 - 30xy + 25y^2 A$

19.

Square of sum

$(a+b)^2=a^2+2ab+b^2$

Polynomial

Power

20.

Cube of difference

$(a-b)^3=a^3-3a^2b+3ab^2-b^3$

Polynomial

Power