Question

Решите тригонометрическое уравнение :
sin(x+30°)=cos(x-30°)
x принадлежит (0 °; 180 °)

Answer 1

$sinleft(x+30^circright)=cosleft(x-30^circright)medskip\sin xcos 30^circ+cos xsin 30^circ=cos xcos 30^circ+sin xsin 30^circmedskip\dfrac{sqrt{3}}{2}sin x+dfrac{1}{2}cos x=dfrac{sqrt{3}}{2}cos x+dfrac{1}{2}sin xmedskip\dfrac{sqrt{3}-1}{2}sin x+dfrac{1-sqrt{3}}{2}cos x=0mid,:cos xneq 0medskip\dfrac{sqrt{3}-1}{2}mathrm{tg},x-dfrac{sqrt{3}-1}{2}=0medskip\mathrm{tg},x=1$

$begin{cases}mathrm{tg},x=1smallskip\xinleft(0^circ,;,180^circright)end{cases}Leftrightarrow x=45^circ$

Ответ. $x=45^circ$