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Popular Trigonometria >

provar tan(-pi/2-θ)=cot(θ)

Solução

provar

tan (- \frac{π}{2} - θ) = cot (θ)

Solução

Verdadeiro

Passos da solução

tan (- \frac{π}{2} - θ) = cot (θ)

Manipular o lado direito

tan (- \frac{π}{2} - θ)

Use a identidade de ângulo negativo:

tan (- x) = - tan (x)

= - tan (\frac{π}{2} + θ)

Reeecreva usando identidades trigonométricas

tan (\frac{π}{2} + θ)

Utilizar a Identidade Básica da Trigonometria:

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

= \frac{sin ( \frac{π}{2} + θ )}{cos ( \frac{π}{2} + θ )}

Use a identidade de soma de ângulos:

sin (s + t) = sin (s) cos (t) + cos (s) sin (t)

= \frac{sin ( \frac{π}{2} ) cos ( θ ) + cos ( \frac{π}{2} ) sin ( θ )}{cos ( \frac{π}{2} + θ )}

Use a identidade de soma de ângulos:

cos (s + t) = cos (s) cos (t) - sin (s) sin (t)

= \frac{sin ( \frac{π}{2} ) cos ( θ ) + cos ( \frac{π}{2} ) sin ( θ )}{cos ( \frac{π}{2} ) cos ( θ ) - sin ( \frac{π}{2} ) sin ( θ )}

Simplificar

\frac{sin ( \frac{π}{2} ) cos ( θ ) + cos ( \frac{π}{2} ) sin ( θ )}{cos ( \frac{π}{2} ) cos ( θ ) - sin ( \frac{π}{2} ) sin ( θ )} : - \frac{cos ( θ )}{sin ( θ )}

\frac{sin ( \frac{π}{2} ) cos ( θ ) + cos ( \frac{π}{2} ) sin ( θ )}{cos ( \frac{π}{2} ) cos ( θ ) - sin ( \frac{π}{2} ) sin ( θ )}

sin (\frac{π}{2}) cos (θ) + cos (\frac{π}{2}) sin (θ) = cos (θ)

sin (\frac{π}{2}) cos (θ) + cos (\frac{π}{2}) sin (θ)

sin (\frac{π}{2}) cos (θ) = cos (θ)

sin (\frac{π}{2}) cos (θ)

Simplificar

sin (\frac{π}{2}) : 1

sin (\frac{π}{2})

Utilizar a seguinte identidade trivial:

sin (\frac{π}{2}) = 1

sin (x)

tabela de periodicidade com ciclo de

2 πn

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

= 1

= 1 \cdot cos (θ)

Multiplicar:

1 \cdot cos (θ) = cos (θ)

= cos (θ)

cos (\frac{π}{2}) sin (θ) = 0

cos (\frac{π}{2}) sin (θ)

Simplificar

cos (\frac{π}{2}) : 0

cos (\frac{π}{2})

Utilizar a seguinte identidade trivial:

cos (\frac{π}{2}) = 0

cos (x)

tabela de periodicidade com ciclo de

2 πn

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

= 0

= 0 \cdot sin (θ)

Aplicar a regra

0 \cdot a = 0

= 0

= cos (θ) + 0

cos (θ) + 0 = cos (θ)

= cos (θ)

= \frac{cos ( θ )}{cos ( \frac{π}{2} ) cos ( θ ) - sin ( \frac{π}{2} ) sin ( θ )}

cos (\frac{π}{2}) cos (θ) - sin (\frac{π}{2}) sin (θ) = - sin (θ)

cos (\frac{π}{2}) cos (θ) - sin (\frac{π}{2}) sin (θ)

cos (\frac{π}{2}) cos (θ) = 0

cos (\frac{π}{2}) cos (θ)

Simplificar

cos (\frac{π}{2}) : 0

cos (\frac{π}{2})

Utilizar a seguinte identidade trivial:

cos (\frac{π}{2}) = 0

cos (x)

tabela de periodicidade com ciclo de

2 πn

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

= 0

= 0 \cdot cos (θ)

Aplicar a regra

0 \cdot a = 0

= 0

sin (\frac{π}{2}) sin (θ) = sin (θ)

sin (\frac{π}{2}) sin (θ)

Simplificar

sin (\frac{π}{2}) : 1

sin (\frac{π}{2})

Utilizar a seguinte identidade trivial:

sin (\frac{π}{2}) = 1

sin (x)

tabela de periodicidade com ciclo de

2 πn

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

= 1

= 1 \cdot sin (θ)

Multiplicar:

1 \cdot sin (θ) = sin (θ)

= sin (θ)

= 0 - sin (θ)

0 - sin (θ) = - sin (θ)

= - sin (θ)

= \frac{cos ( θ )}{- sin ( θ )}

Aplicar as propriedades das frações:

\frac{a}{- b} = - \frac{a}{b}

= - \frac{cos ( θ )}{sin ( θ )}

= - \frac{cos ( θ )}{sin ( θ )}

= - (- \frac{cos ( θ )}{sin ( θ )})

Aplicar a regra

- (- a) = a

= \frac{cos ( θ )}{sin ( θ )}

Utilizar a Identidade Básica da Trigonometria:

\frac{cos ( x )}{sin ( x )} = cot (x)

= cot (θ)

Demonstramos que os dois lados podem adquirir a mesma forma

\Rightarrow Verdadeiro

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