Answer to Question #257369 in Algebra for saduni

Question

Using Euler’s formular obtain trigonometric formulars for cos(𝜃1 + 𝜃2) and sin(𝜃1 + 𝜃2).

Answer 1

Formulas;

$cos (\theta_1+\theta_2) and \space sin(\theta_1+\theta_2)$

$e^{i(\theta_1+\theta_2)}=e^{i\theta_1+i\theta_2}=e^{i\theta_1}.e^{i\theta_2}$

$cos(\theta_1+\theta_2)+isin(\theta_1+\theta_2)=(cos\theta_1+isin\theta_1)(cos\theta_2+isin\theta_2)$

$=cos\theta_1cos\theta_2-sin\theta_1sin\theta_2+(sin\theta_1cos\theta_2+cos\theta_1sin\theta_2)i$

Equate real and imaginary parts to get that;

$cos(\theta_1+\theta_2)=cos\theta_1cos\theta_2-sin\theta_1sin\theta_2$

$sin(\theta_1+\theta_2)=sin\theta_1cos\theta_2+cos\theta_1sin\theta_2$