Trigonometria Exemplos

sin (\frac{π}{4} + x)

$\sin (\frac{π}{4} + x)$

Etapa 1

Use a fórmula da soma do seno para simplificar a expressão. A fórmula determina que

sin (A + B) = sin (A) cos (B) + cos (A) sin (B)

$\sin (A + B) = \sin (A) \cos (B) + \cos (A) \sin (B)$ .

sin (\frac{π}{4}) cos (x) + cos (\frac{π}{4}) sin (x)

$\sin (\frac{π}{4}) \cos (x) + \cos (\frac{π}{4}) \sin (x)$

Etapa 2

Remova os parênteses.

sin (\frac{π}{4}) cos (x) + cos (\frac{π}{4}) sin (x)

$\sin (\frac{π}{4}) \cos (x) + \cos (\frac{π}{4}) \sin (x)$

Etapa 3

Simplifique cada termo.

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Etapa 3.1

O valor exato de

sin (\frac{π}{4})

$\sin (\frac{π}{4})$ é

\frac{\sqrt{2}}{2}

$\frac{\sqrt{2}}{2}$ .

\frac{\sqrt{2}}{2} cos (x) + cos (\frac{π}{4}) sin (x)

$\frac{\sqrt{2}}{2} \cos (x) + \cos (\frac{π}{4}) \sin (x)$

Etapa 3.2

Combine

\frac{\sqrt{2}}{2}

$\frac{\sqrt{2}}{2}$ e

cos (x)

$\cos (x)$ .

\frac{\sqrt{2} cos (x)}{2} + cos (\frac{π}{4}) sin (x)

$\frac{\sqrt{2} \cos (x)}{2} + \cos (\frac{π}{4}) \sin (x)$

Etapa 3.3

O valor exato de

cos (\frac{π}{4})

$\cos (\frac{π}{4})$ é

\frac{\sqrt{2}}{2}

$\frac{\sqrt{2}}{2}$ .

\frac{\sqrt{2} cos (x)}{2} + \frac{\sqrt{2}}{2} sin (x)

$\frac{\sqrt{2} \cos (x)}{2} + \frac{\sqrt{2}}{2} \sin (x)$

Etapa 3.4

Combine

\frac{\sqrt{2}}{2}

$\frac{\sqrt{2}}{2}$ e

sin (x)

$\sin (x)$ .

\frac{\sqrt{2} cos (x)}{2} + \frac{\sqrt{2} sin (x)}{2}

$\frac{\sqrt{2} \cos (x)}{2} + \frac{\sqrt{2} \sin (x)}{2}$

\frac{\sqrt{2} cos (x)}{2} + \frac{\sqrt{2} sin (x)}{2}

$\frac{\sqrt{2} \cos (x)}{2} + \frac{\sqrt{2} \sin (x)}{2}$

Etapa 4

Simplifique.

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Etapa 4.1

Combine os numeradores em relação ao denominador comum.

\frac{\sqrt{2} cos (x) + \sqrt{2} sin (x)}{2}

$\frac{\sqrt{2} \cos (x) + \sqrt{2} \sin (x)}{2}$

Etapa 4.2

Fatore

\sqrt{2}

$\sqrt{2}$ de

\sqrt{2} cos (x) + \sqrt{2} sin (x)

$\sqrt{2} \cos (x) + \sqrt{2} \sin (x)$ .

\frac{\sqrt{2} (cos (x) + sin (x))}{2}

$\frac{\sqrt{2} (\cos (x) + \sin (x))}{2}$

\frac{\sqrt{2} (cos (x) + sin (x))}{2}

$\frac{\sqrt{2} (\cos (x) + \sin (x))}{2}$