Trigonometria Exemplos

$\cos (2 π t - \frac{π}{4})$

Etapa 1

Aplique a fórmula da diferença dos ângulos $\cos (x - y) = \cos (x) \cos (y) + \sin (x) \sin (y)$ .

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

Etapa 2

Simplifique cada termo.

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Etapa 2.1

Adicione parênteses.

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

Etapa 2.2

Use a fórmula do arco duplo para transformar $\cos (2 x)$ em $\cos^{2} (x) - \sin^{2} (x)$ .

$(\cos^{2} (π t) - \sin^{2} (π t)) \cos (\frac{π}{4}) + \sin (2 π t) \sin (\frac{π}{4})$

Etapa 2.3

O valor exato de $\cos (\frac{π}{4})$ é $\frac{\sqrt{2}}{2}$ .

$(\cos^{2} (π t) - \sin^{2} (π t)) \frac{\sqrt{2}}{2} + \sin (2 π t) \sin (\frac{π}{4})$

Etapa 2.4

Aplique a propriedade distributiva.

$\cos^{2} (π t) \frac{\sqrt{2}}{2} - \sin^{2} (π t) \frac{\sqrt{2}}{2} + \sin (2 π t) \sin (\frac{π}{4})$

Etapa 2.5

Combine $\cos^{2} (π t)$ e $\frac{\sqrt{2}}{2}$ .

$\frac{\cos^{2} (π t) \sqrt{2}}{2} - \sin^{2} (π t) \frac{\sqrt{2}}{2} + \sin (2 π t) \sin (\frac{π}{4})$

Etapa 2.6

Combine $\frac{\sqrt{2}}{2}$ e $\sin^{2} (π t)$ .

$\frac{\cos^{2} (π t) \sqrt{2}}{2} - \frac{\sqrt{2} \sin^{2} (π t)}{2} + \sin (2 π t) \sin (\frac{π}{4})$

Etapa 2.7

Adicione parênteses.

$\frac{\cos^{2} (π t) \sqrt{2}}{2} - \frac{\sqrt{2} \sin^{2} (π t)}{2} + \sin (2 (π t)) \sin (\frac{π}{4})$

Etapa 2.8

Aplique a fórmula do arco duplo do seno.

$\frac{\cos^{2} (π t) \sqrt{2}}{2} - \frac{\sqrt{2} \sin^{2} (π t)}{2} + 2 \sin (π t) \cos (π t) \sin (\frac{π}{4})$

Etapa 2.9

O valor exato de $\sin (\frac{π}{4})$ é $\frac{\sqrt{2}}{2}$ .

$\frac{\cos^{2} (π t) \sqrt{2}}{2} - \frac{\sqrt{2} \sin^{2} (π t)}{2} + 2 \sin (π t) \cos (π t) \frac{\sqrt{2}}{2}$

Etapa 2.10

Cancele o fator comum de $2$ .

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Etapa 2.10.1

Fatore $2$ de $2 \sin (π t) \cos (π t)$ .

$\frac{\cos^{2} (π t) \sqrt{2}}{2} - \frac{\sqrt{2} \sin^{2} (π t)}{2} + 2 (\sin (π t) \cos (π t)) \frac{\sqrt{2}}{2}$

Etapa 2.10.2

Cancele o fator comum.

$\frac{\cos^{2} (π t) \sqrt{2}}{2} - \frac{\sqrt{2} \sin^{2} (π t)}{2} + 2 (\sin (π t) \cos (π t)) \frac{\sqrt{2}}{2}$

Etapa 2.10.3

Reescreva a expressão.

$\frac{\cos^{2} (π t) \sqrt{2}}{2} - \frac{\sqrt{2} \sin^{2} (π t)}{2} + \sin (π t) \cos (π t) \sqrt{2}$