Question

Упростите выражение:cos(30°+a)-sin(60°+a)/cos(30°-a)-sin(60°-a)​

Answer 1

Ответ:

Объяснение:

(sin30cosa+sinacos30-cos60cosa+sin60sina)

(sin30cosa+sinacos30+cos60cosa-sin60sina)=(1/2*cosa+√3/2*sina-1/2*cosa+√3/2*sina)/ (1/2*cosa+√3/2*sina+1/2*cosa-√3/2*sina)=

=√3sina/cosa=√3tga

Answer 2

Давайте упростим данное выражение по шагам.

У нас есть выражение:

\[ \frac{\cos(30^\circ + a) - \sin(60^\circ + a)}{\cos(30^\circ - a) - \sin(60^\circ - a)} \]

Шаг 1: Воспользуемся тригонометрическими формулами сложения и вычитания для косинуса и синуса.

\[ \cos(30^\circ + a) = \cos(30^\circ)\cos(a) - \sin(30^\circ)\sin(a) \]

\[ \sin(60^\circ + a) = \sin(60^\circ)\cos(a) + \cos(60^\circ)\sin(a) \]

\[ \cos(30^\circ - a) = \cos(30^\circ)\cos(a) + \sin(30^\circ)\sin(a) \]

\[ \sin(60^\circ - a) = \sin(60^\circ)\cos(a) - \cos(60^\circ)\sin(a) \]

Подставим эти значения в наше исходное выражение:

\[ \frac{(\cos(30^\circ)\cos(a) - \sin(30^\circ)\sin(a)) - (\sin(60^\circ)\cos(a) + \cos(60^\circ)\sin(a))}{(\cos(30^\circ)\cos(a) + \sin(30^\circ)\sin(a)) - (\sin(60^\circ)\cos(a) - \cos(60^\circ)\sin(a))} \]

Шаг 2: Сгруппируем слагаемые по типу:

\[ \frac{\cos(30^\circ)\cos(a) - \sin(30^\circ)\sin(a) - \sin(60^\circ)\cos(a) - \cos(60^\circ)\sin(a)}{\cos(30^\circ)\cos(a) + \sin(30^\circ)\sin(a) - \sin(60^\circ)\cos(a) + \cos(60^\circ)\sin(a)} \]

Шаг 3: Сложим и вынесем общий множитель в числителе и знаменателе:

\[ \frac{(\cos(30^\circ)\cos(a) - \sin(30^\circ)\sin(a)) - (\sin(60^\circ)\cos(a) + \cos(60^\circ)\sin(a))}{(\cos(30^\circ)\cos(a) + \sin(30^\circ)\sin(a)) - (\sin(60^\circ)\cos(a) - \cos(60^\circ)\sin(a))} \]

\[ = \frac{\cos(30^\circ)\cos(a) - \sin(30^\circ)\sin(a) - \sin(60^\circ)\cos(a) - \cos(60^\circ)\sin(a)}{\cos(30^\circ)\cos(a) + \sin(30^\circ)\sin(a) - \sin(60^\circ)\cos(a) + \cos(60^\circ)\sin(a)} \]

\[ = \frac{\cos(30^\circ)\cos(a) - \sin(30^\circ)\sin(a) - \sin(60^\circ)\cos(a) - \cos(60^\circ)\sin(a)}{\cos(30^\circ)\cos(a) + \sin(30^\circ)\sin(a) - \sin(60^\circ)\cos(a) + \cos(60^\circ)\sin(a)} \]

\[ = \frac{\cos(30^\circ)\cos(a) - \sin(30^\circ)\sin(a) - \sin(60^\circ)\cos(a) - \cos(60^\circ)\sin(a)}{\cos(30^\circ)\cos(a) + \sin(30^\circ)\sin(a) - \sin(60^\circ)\cos(a) + \cos(60^\circ)\sin(a)} \]

\[ = \frac{\cos(30^\circ)\cos(a) - \sin(30^\circ)\sin(a) - \sin(60^\circ)\cos(a) - \cos(60^\circ)\sin(a)}{\cos(30^\circ)\cos(a) + \sin(30^\circ)\sin(a) - \sin(60^\circ)\cos(a) + \cos(60^\circ)\sin(a)} \]

Шаг 4: Упростим числитель и знаменатель:

\[ = \frac{\cos(30^\circ)\cos(a) - \sin(30^\circ)\sin(a) - \sin(60^\circ)\cos(a) - \cos(60^\circ)\sin(a)}{\cos(30^\circ)\cos(a) + \sin(30^\circ)\sin(a) - \sin(60^\circ)\cos(a) + \cos(60^\circ)\sin(a)} \]

\[ = \frac{\cos(30^\circ)\cos(a) - \sin(30^\circ)\sin(a) - \sqrt{3}\cos(a) + \frac{1}{2}\sin(a)}{\cos(30^\circ)\cos(a) + \sin(30^\circ)\sin(a) - \frac{\sqrt{3}}{2}\cos(a) + \frac{1}{2}\sin(a)} \]

Шаг 5: Упростим дальше, объединив слагаемые:

\[ = \frac{\left(\cos(30^\circ) - \sqrt{3}\right)\cos(a) - \left(\sin(30^\circ) - \frac{1}{2}\right)\sin(a)}{\left(\cos(30^\circ) + \frac{\sqrt{3}}{2}\right)\cos(a) + \left(\sin(30^\circ) + \frac{1}{2}\right)\sin(a)} \]

Таким образом, упрощенное выражение для \(\frac{\cos(30^\circ + a) - \sin(60^\circ + a)}{\cos(30^\circ - a) - \sin(60^\circ - a)}\) равно \(\frac{\left(\cos(30^\circ) - \sqrt{3}\right)\cos(a) - \left(\sin(30^\circ) - \frac{1}{2}\right)\sin(a)}{\left(\cos(30^\circ) + \frac{\sqrt{3}}{2}\right)\cos(a) + \left(\sin(30^\circ) + \frac{1}{2}\right)\sin(a)}\).