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provar sin(θ)+cos(θ)=sqrt(2)sin(θ+45)

Solução

provar

sin (θ) + cos (θ) = 2 sin (θ + 4 5^{\circ})

Verdadeiro

Passos da solução

sin (θ) + cos (θ) = 2 sin (θ + 4 5^{\circ})

Manipular o lado esquerdo

2 sin (θ + 4 5^{\circ})

Reeecreva usando identidades trigonométricas

sin (θ + 4 5^{\circ})

Use a identidade de soma de ângulos:

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

= sin (θ) cos (4 5^{\circ}) + cos (θ) sin (4 5^{\circ})

Simplificar

sin (θ) cos (4 5^{\circ}) + cos (θ) sin (4 5^{\circ}) : \frac{2 sin ( θ ) + 2 cos ( θ )}{2}

sin (θ) cos (4 5^{\circ}) + cos (θ) sin (4 5^{\circ})

sin (θ) cos (4 5^{\circ}) = \frac{2 sin ( θ )}{2}

sin (θ) cos (4 5^{\circ})

Simplificar

cos (4 5^{\circ}) : \frac{2}{2}

cos (4 5^{\circ})

Utilizar a seguinte identidade trivial:

cos (4 5^{\circ}) = \frac{2}{2}

cos (x)

tabela de periodicidade com ciclo de

36 0^{\circ} n

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

= \frac{2}{2}

= \frac{2}{2} sin (θ)

Multiplicar frações:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{2 sin ( θ )}{2}

cos (θ) sin (4 5^{\circ}) = \frac{2 cos ( θ )}{2}

cos (θ) sin (4 5^{\circ})

Simplificar

sin (4 5^{\circ}) : \frac{2}{2}

sin (4 5^{\circ})

Utilizar a seguinte identidade trivial:

sin (4 5^{\circ}) = \frac{2}{2}

sin (x)

tabela de periodicidade com ciclo de

36 0^{\circ} n

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

= \frac{2}{2}

= \frac{2}{2} cos (θ)

Multiplicar frações:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{2 cos ( θ )}{2}

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

Aplicar a regra

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{2 sin ( θ ) + 2 cos ( θ )}{2}

= \frac{2 sin ( θ ) + 2 cos ( θ )}{2}

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

Simplificar

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

2 \frac{2 sin ( θ ) + 2 cos ( θ )}{2}

Multiplicar frações:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

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

Cancelar

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

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

Aplicar as propriedades dos radicais:

n a = a^{\frac{1}{n}}

2 = 2^{\frac{1}{2}}

= \frac{2 ^{\frac{1}{2}} ( 2 sin ( θ ) + 2 cos ( θ ) )}{2}

Aplicar as propriedades dos expoentes:

\frac{x ^{a}}{x ^{b}} = \frac{1}{x ^{b - a}}

\frac{2 ^{\frac{1}{2}}}{2 ^{1}} = \frac{1}{2 ^{1 - \frac{1}{2}}}

= \frac{2 sin ( θ ) + 2 cos ( θ )}{2 ^{1 - \frac{1}{2}}}

Subtrair:

1 - \frac{1}{2} = \frac{1}{2}

= \frac{2 sin ( θ ) + 2 cos ( θ )}{2 ^{\frac{1}{2}}}

Aplicar as propriedades dos radicais:

a^{\frac{1}{n}} = n a

2^{\frac{1}{2}} = 2

= \frac{2 sin ( θ ) + 2 cos ( θ )}{2}

= \frac{2 sin ( θ ) + 2 cos ( θ )}{2}

Fatorar o termo comum

2

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

Eliminar o fator comum:

2

= sin (θ) + cos (θ)

= sin (θ) + cos (θ)

Demonstramos que os dois lados podem adquirir a mesma forma

\Rightarrow Verdadeiro

p ro v e \frac{1 - tan ( θ )}{1 + tan ( θ )} = \frac{cot ( θ ) - 1}{cot ( θ ) + 1}

p ro v e cot (θ) sin (θ) cos (θ) = 1 - sin^{2} (θ)

p ro v e csc (θ) = \frac{1}{sin ( θ )}

p ro v e \frac{1}{sin ( θ )} - sin (θ) = cot (θ) cos (θ)

p ro v e \frac{sec ( - x )}{tan ( x ) + cot ( x )} = - sin (- x)