Tabelas e relações particulares

Transformada de Fourier

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Relações Trigonométricas Usuais

sin²x + cos²x = 1        cos²x = $${{1+\cos 2x}\over 2}$$        sin²x = $${{1-\cos 2x}\over 2}$$

mathend000#

sin x = $${{e^{jx} - e^{-jx}}\over 2j}$$        cos x = $${{e^{jx} + e^{-jx}}\over 2}$$        tan x = $${{e^{jx} - e^{-jx}}\over {j(e^{jx} + e^{-jx})}}$$

mathend000#

Adição

sin(a + b) = sin a cos b + sin b cos a        sin(a - b) = sin a cos b - sin b cos a

mathend000#

cos(a + b) = cos a cos b - sin a sin b        cos(a - b) = cos a cos b + sin a sin b

mathend000#

tan(a + b) = $${{\tan a + \tan b}\over {1-\tan a \tan b}}$$        tan(a - b) = $${{\tan a - \tan b}\over {1+\tan a \tan b}}$$

mathend000#

Multiplicação: com tan a = t mathend000#

sin(2a) = 2 sin a cos a = $${{2t}\over {1+t^2}}$$

mathend000#

cos(2a) = cos²a - sin²a = 2 cos²a - 1 = 1 - 2 sin²a = $${{1-t^2}\over {1+t^2}}$$

mathend000#

tan(2a) = $${{2\tan a}\over {1-\tan^2 a}}$$ = $${{2t}\over {1-t^2}}$$

mathend000#

cos a cos b = $${1\over 2}$$[cos(a + b) + cos(a - b)]        sin a sin b = $${1\over 2}$$[cos(a - b) - cos(a + b)]

mathend000#

sin a cos b = $${1\over 2}$$[sin(a + b) + sin(a - b)]

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cos p + cos q = 2 cos$${{p+q}\over 2}$$cos$${{p-q}\over 2}$$        cos p - cos q = - 2 sin$${{p+q}\over 2}$$sin$${{p-q}\over 2}$$

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sin p + sin q = 2 sin$${{p+q}\over 2}$$cos$${{p-q}\over 2}$$        sin p - sin q = 2 sin$${{p-q}\over 2}$$cos$${{p+q}\over 2}$$

mathend000#

tan p + tan q = $${{\sin(p+q)}\over {\cos p \cos q}}$$        tan p - tan q = $${{\sin(p-q)}\over {\cos p \cos q}}$$

mathend000#

Trigonometria Hiperbólica

cosh x + sinh x = exp(x)        cosh x - sinh x = exp(- x)        cosh²x - sinh²x = 1

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sinh(a + b) = sinh a cosh b + sinh b cosh a        sinh(a - b) = sinh a cosh b - sinh b cosh a

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cosh(a + b) = cosh a cosh b + sinh a sinh b        cosh(a - b) = cosh a cosh b - sinh a sinh b

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sinh 2a = 2 sinh a cosh a        cosh 2a = cosh²a sinh²a = 1 + 2 sin²a = 2 cosh²a - 1

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cosh²a = $${{1+\cosh 2a}\over 2}$$        sinh²a = $${{\cosh 2a -1}\over 2}$$        tanh²a = $${{\cosh 2a -1}\over {\cosh 2a +1}}$$

mathend000#

tanh 2a = $${{2\tanh a}\over {1+\tanh^2 a}}$$

mathend000#

sinh x = $${{e^{x} - e^{-x}}\over 2}$$        cosh = $${{e^{x} + e^{-x}}\over 2}$$        tanh = $${{e^{x} - e^{-x}}\over {e^{x} + e^{-x}}}$$

mathend000#

cosh jx = cos x        sinh jx = j sin x        tanh jx = j tan x

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Desenvolvimentos em série

sin x = x - ${{x^3}\over {3!}}$ + ${{x^5}\over {5!}}$ +...+ (- 1)ⁿ${{x^{2n+1}}\over {(2n+1)!}}$ + x²ⁿ⁺¹$\epsilon$(x) mathend000#

cos x = 1 - ${{x^2}\over {2!}}$ + ${{x^4}\over {4!}}$ +...+ (- 1)ⁿ${{x^{2n}}\over {(2n)!}}$ + x²ⁿ$\epsilon$(x) mathend000#

tan x = x - ${{x^3}\over {3}}$ + ${{2x^5}\over {15}}$ + x⁵$\epsilon$(x) mathend000#

sinh x = x + ${{x^3}\over {3!}}$ +...+ ${{x^{2n-1}}\over {(2n-1)!}}$ + x^2n-1$\epsilon$(x) mathend000#

cosh x = 1 + ${{x^2}\over {2!}}$ +...+ ${{x^{2n}}\over {(2n)!}}$ + x²ⁿ$\epsilon$(x) mathend000#

tanh x = x - ${{x^3}\over {3}}$ + ${{2x^5}\over {15}}$ + x⁵$\epsilon$(x) mathend000#

arcsin x = x + ${{x^3}\over {6}}$ +...+ ${{1.3.5\ldots(2n+1)}\over {2.4.6\ldots(2n)}}$${{x^{2n+1}}\over {(2n+1)!}}$ - x²ⁿ⁺¹$\epsilon$(x) mathend000#

arctan x = x - ${{x^3}\over {3}}$ + ${{x^5}\over {5}}$ +...+ (- 1)ⁿ${{x^{2n+1}}\over {(2n+1)!}}$ + x²ⁿ⁺¹$\epsilon$(x) mathend000#

sinh^-1x = x - ${{x^3}\over {6}}$ +...+ (- 1)ⁿ${{1.3.5\ldots(2n+1)}\over {2.4.6\ldots(2n)}}$${{x^{2n+1}}\over {(2n+1)!}}$ + x²ⁿ⁺¹$\epsilon$(x) mathend000#

tanh^-1x = x + ${{x^3}\over {3}}$ + ${{x^5}\over 5}$ +...+ ${{x^{2n+1}}\over {(2n+1)!}}$ + x²ⁿ⁺¹$\epsilon$(x) mathend000#

e^x = 1 + x + ${{x^2}\over {2!}}$ + ${{x^3}\over {3!}}$ +...+ ${{x^n}\over {n!}}$ + xⁿ$\epsilon$(x) mathend000#

log(1 + x) = x - ${{x^2}\over {2}}$ + ${{x^3}\over {3}}$ +...+ (- 1)^n-1${{x^n}\over {n}}$ + xⁿ$\epsilon$(x) mathend000#

(1 + x)^$\alpha$ = 1 + $\alpha$x + $\alpha$($\alpha$ -1)${{x^2}\over {2}}$ +...+ $\alpha$($\alpha$ -1)...($\alpha$ - n + 1)${{x^n}\over {n}}$ + xⁿ$\epsilon$(x) mathend000#

${1\over {1-x}}$ = 1 + x + x² +...+ xⁿ + xⁿ$\epsilon$(x) mathend000#

${1\over {1+x}}$ = 1 - x + x² - x³ +...+ (- 1)ⁿxⁿ + xⁿ$\epsilon$(x) mathend000#

Algumas relações úteis

Integrais

$\int_{0}^{{\infty}}$e^-ax2dx = ${1\over 2}$$\sqrt{{\pi\over a}}$ mathend000#

$\int_{0}^{{\infty}}$xe^-ax2dx = ${1\over 2a}$ mathend000#

$\int_{0}^{{\infty}}$x²e^-ax2dx = ${{\sqrt{\pi}}\over {4a^{3/2}}}$ mathend000#

$\int_{0}^{{\infty}}$x³e^-ax2dx = ${1\over {2a^2}}$ mathend000#

$\int_{0}^{{\infty}}$x⁴e^-ax2dx = ${3\over {8a^2}}$$\sqrt{{\pi\over a}}$ mathend000#

Séries

Geométrica: u₁ + qu₁ + q²u₁ +...+ q^n-1u₁ = u₁${{1-q^n}\over {1-q}}$ mathend000#

Aritmética: u₁ + qu₁ +2qu₁ +...+ (n - 1)qu₁ = mathend000#

Derivadas

[a^f(x)]' = log aa^f(x)f'(x) mathend000#

Trigonometria do círculo

	0	$\pi$$\over$6 mathend000#	$\pi$$\over$4 mathend000#	$\pi$$\over$3 mathend000#	$\pi$$\over$2 mathend000#
sin x mathend000#	0	1/2	$\sqrt{{2}}$/2 mathend000#	$\sqrt{{3}}$/2 mathend000#	1
cos x mathend000#	1	$\sqrt{{3}}$/2 mathend000#	$\sqrt{{2}}$/2 mathend000#	1/2	0
tan x mathend000#	0	$\sqrt{{3}}$/3 mathend000#	1	$\sqrt{{3}}$ mathend000#	$\infty$ mathend000#
cot x mathend000#	$\infty$ mathend000#	$\sqrt{{3}}$ mathend000#	1	$\sqrt{{3}}$/3 mathend000#	0