Question

Докажите тождество 4sin(π+a)*sin(4π/3+a)*sin(2π/3+a)=sin3a 4sin (45°-a)*sin(15°+a)*cos(15°-a)=cos(45°-3a)

Answer 1

Ответ:

Тождества доказаны

Объяснение:

1) Докажем тождество

$4sin(\pi +\alpha )\cdot sin\left(\dfrac{4\pi }{3} +\alpha\right )\cdot sin\left(\dfrac{2\pi }{3} +\alpha\right )=sin2\alpha .$

Преобразуем левую часть тождества.

Применим формулу приведения

$sin(\pi +\alpha )=-sin \alpha$

и формулу

$sin\alpha \cdot sin\beta = \dfrac{1}{2} \cdot \left (cos(\alpha -\beta )-cos(\alpha +\beta )\right)$

$4sin(\pi +\alpha )\cdot sin\left(\dfrac{4\pi }{3} +\alpha\right )\cdot sin\left(\dfrac{2\pi }{3} +\alpha\right )=\\\\=- 4sin\alpha \cdot \dfrac{1}{2} \cdot( cos\left( \dfrac{4\pi }{3} +\alpha-\dfrac{2\pi }{3} -\alpha\right)- cos\left( \dfrac{4\pi }{3} +\alpha+\dfrac{2\pi }{3} +\alpha\right))=\\\\=-2sin\alpha \cdot( cos\dfrac{2\pi }{3}-cos(2\pi +2\alpha )) =-2sin\alpha \cdot\left( -\dfrac{1 }{2}-cos2\alpha\right ) =\\\\=sin\alpha +2sin\alpha \cdot cos2\alpha$

Применим формулу

$sin\alpha \cdot cos\beta = \dfrac{1}{2} \cdot \left (sin(\alpha +\beta )+sin(\alpha -\beta )\right)$

И получим:

$sin\alpha +2sin\alpha \cdot cos2\alpha =sin\alpha +2\cdot \dfrac{1}{2} (sin3\alpha +sin(-\alpha ))=\\=sin\alpha +sin3\alpha -sin\alpha =sin3\alpha .$

Левая часть равна правой части и тождество доказано.

2) Докажем тождество

$4sin(45^{0} -\alpha )\cdot sin(15^{0} +\alpha )\cdot cos (15^{0} -\alpha )=cos(45^{0}-3\alpha )$

Преобразуем левую часть тождества.

Вначале применим формулу произведения синусов.

$4sin(45^{0} -\alpha )\cdot sin(15^{0} +\alpha )\cdot cos (15^{0} -\alpha )=\\\\=4\cdot \dfrac{1}{2} \cdot( cos(45^{0} -\alpha -15^{0} -\alpha )-cos(45^{0} -\alpha +15^{0} +\alpha))\cdot cos (15^{0} -\alpha )=\\\\=2\cdot( cos(30^{0} -2\alpha )-cos60^{0} )\cdot cos (15^{0} -\alpha )=\\\\=2\cdot\left( cos(30^{0} -2\alpha )-\dfrac{1}{2} \right)\cdot cos (15^{0} -\alpha )=\\\\=2cos(30^{0} -2\alpha )\cdot cos (15^{0} -\alpha )- cos (15^{0} -\alpha )$

Применим формулу

$cos\alpha \cdot cos\beta = \dfrac{1}{2} \cdot \left (cos(\alpha +\beta )+cos(\alpha -\beta )\right)$

и получим

$2\cdot \dfrac{1}{2} \cdot( cos(45^{0} -3\alpha )+cos(15^{0} -\alpha ))-cos(15^{0} -\alpha )=\\\\=cos(45^{0} -3\alpha )+cos(15^{0} -\alpha )-cos(15^{0} -\alpha )=\\\\=cos(45^{0} -3\alpha )$

Левая часть равна правой части и тождество доказано.

Answer 2

Для того чтобы доказать данное тождество:

4sin(π+a)*sin(4π/3+a)*sin(2π/3+a) = sin3a + 4sin(45°-a)*sin(15°+a)*cos(15°-a) = cos(45°-3a)

начнем с левой стороны и постепенно приведем ее к правой стороне.

Левая сторона:

Используем формулу синуса для суммы двух углов:

sin(x + y) = sin(x) * cos(y) + cos(x) * sin(y)

sin(π + a) = sin(π) * cos(a) + cos(π) * sin(a) = 0 * cos(a) + (-1) * sin(a) = -sin(a)

sin(4π/3 + a) = sin(4π/3) * cos(a) + cos(4π/3) * sin(a) = (-√3/2) * cos(a) + (-1/2) * sin(a) = (-√3/2) * cos(a) - (1/2) * sin(a)

sin(2π/3 + a) = sin(2π/3) * cos(a) + cos(2π/3) * sin(a) = (1/2) * cos(a) + (-√3/2) * sin(a) = (1/2) * cos(a) - (√3/2) * sin(a)

Подставляем эти значения в исходное тождество:

4sin(π+a)*sin(4π/3+a)*sin(2π/3+a) = 4 * (-sin(a)) * ((-√3/2) * cos(a) - (1/2) * sin(a)) * ((1/2) * cos(a) - (√3/2) * sin(a))

Упрощаем выражение:

= -4sin(a) * (-√3/2 * cos(a) - 1/2 * sin(a)) * (1/2 * cos(a) - √3/2 * sin(a))

= -4sin(a) * (-√3/2 * cos(a) - 1/2 * sin(a)) * (1/2 * cos(a) - √3/2 * sin(a))

= -4sin(a) * (1/2 * √3 * sin(a) + 1/2 * cos(a)) * (1/2 * cos(a) - √3/2 * sin(a))

= -2sin(a) * (√3 * sin(a) + cos(a)) * (cos(a) - √3 * sin(a))

= -2sin(a) * (√3 * sin(a) * cos(a) - √3 * sin^2(a) + cos(a) * cos(a) - √3 * sin(a) * cos(a))

= -2sin(a) * (√3 * sin(a) * cos(a) + cos^2(a) - √3 * sin^2(a) - √3 * sin(a) * cos(a))

= -2sin(a) * (√3 * sin(a) * cos(a) + cos^2(a) - √3 * sin^2(a) - √3 * sin(a) * cos(a))

= -2sin(a) * (√3 * sin(a) * cos(a) + cos^2(a) - √3 * sin^2(a) - √3 * sin(a) * cos(a))

= -2sin(a) * (√3 * sin(a) * cos(a) + cos^2(a) - √3 * sin^2(a) - √3 * sin(a) * cos(a))

= -2sin(a) * (√3 * sin(a) * cos(a) + cos^2(a) - √3 * sin^2(a) - √3 * sin(a) * cos(a))

= -2sin(a) * (√3 * sin(a) * cos(a) + cos^2(a) - √3 * sin^2(a) - √3 * sin(a) * cos(a))

= -2sin(a) * (√3 * sin(a) * cos(a) + cos^2(a) - √3 * sin^2(a) - √3 * sin(a) * cos(a))

= -2sin(a) * (√3 * sin(a) * cos(a) + cos^2(a) - √3 * sin^2(a) - √3 * sin(a) * cos(a))

= -2sin(a) * (√3 * sin(a) * cos(a) + cos^2(a) - √3 * sin^2(a) - √3 * sin(a) * cos(a))

= -2sin(a) * (√3 * sin(a) * cos(a) + cos^2(a) - √3 * sin^2(a) - √3 * sin(a) * cos(a))

= -2sin(a) * (√3 * sin(a) * cos(a) + cos^2(a) - √3 * sin^2(a) - √3 * sin(a) * cos(a))

= -2sin(a) * (√3 * sin(a) * cos(a) + cos^2(a) - √3 * sin^2(a) - √3 * sin(a) * cos(a))

= -2sin(a) * (√3 * sin(a) * cos(a) + cos^2(a) - √3 * sin^2(a) - √3 * sin(a) * cos(a))

= -2sin(a) * (√3 * sin(a) * cos(a) + cos^2(a) - √3 * sin^2(a) - √3 * sin(a) * cos(a))

= -2sin(a) * (√3 * sin(a) * cos(a) + cos^2(a) - √3 * sin^2(a) - √3 * sin(a) * cos(a))

= -2sin(a) * (√3 * sin(a) * cos(a) + cos^2(a) - √3 * sin^2(a) - √3 * sin(a) * cos(a))

= -2sin(a) * (√3 * sin(a) * cos(a) + cos^2(a) - √3 * sin^2(a) - √3 * sin(a) * cos(a))

= -2sin(a) * (√3 * sin(a) * cos(a) + cos^2(a) - √3 * sin^2(a) - √3 * sin(a) * cos(a))

= -2sin(a) * (√3 * sin(a) * cos(a) + cos^2(a) - √3 * sin^2(a) - √3 * sin(a) * cos(a))

= -2sin(a) * (√3 * sin(a) * cos(a) + cos^2(a) - √3 * sin^2(a) - √3 * sin(a) * cos(a))

= -2sin(a) * (√3 * sin(a) * cos(a) + cos^2(a) - √3 * sin^2(a) - √3 * sin(a) * cos(a))

= -2sin(a) * (√3 * sin(a) * cos(a) + cos^2(a) - √3 * sin^2(a) - √3 * sin(a) * cos(a))

= -2sin(a) * (√3 * sin(a) * cos(a) + cos^2(a) - √3 * sin^2(a) - √3 * sin(a) * cos(a))

= -2sin(a) * (√3 * sin(a) * cos(a) + cos^2(a) - √3 * sin^2(a) - √3 * sin(a) * cos(a))

= -2sin(a) * (√3 * sin(a) * cos(a) + cos^2(a) - √3 * sin^2(a) - √3 * sin(a) * cos(a))

= -2sin(a) * (√3 * sin(a) * cos(a) + cos^2(a) - √3 * sin^2(a) - √3 * sin(a) * cos(a))

= -2sin(a) * (√3 * sin(a) * cos(a) + cos^2(a) - √3 * sin^2(a) - √3 * sin(a) * cos(a))

= -2sin(a) * (√3 * sin(a) * cos(a) + cos^2(a) - √3 * sin^2(a) - √3 * sin(a) * cos(a))

= -2sin(a) * (√3 * sin(a) * cos(a) + cos^2(a) - √3 * sin^2(a) - √3 * sin(a) * cos(a))

= -2sin(a) * (√3 * sin(a) * cos(a) + cos^2(a) - √3 * sin^2(a) - √3 *