1+sin(a)-cos(2*a)-sin(3*a)*1/2*sin(a)^2+sin(a)-1 если a=1/4 (упростите выражение)

Дано

$$\sin{\left (a \right )} + 1 — \cos{\left (2 a \right )} — \frac{1}{2} \sin^{2}{\left (a \right )} \sin{\left (3 a \right )} + \sin{\left (a \right )} — 1$$
Подстановка условия
$$\sin{\left (a \right )} + 1 — \cos{\left (2 a \right )} — \frac{1}{2} \sin^{2}{\left (a \right )} \sin{\left (3 a \right )} + \sin{\left (a \right )} — 1$$

1 + sin((1/4)) — cos(2*(1/4)) — sin(3*(1/4))/2*sin((1/4))^2 + sin((1/4)) — 1

$$\sin{\left ((1/4) \right )} + 1 — \cos{\left (2 (1/4) \right )} — \frac{1}{2} \sin^{2}{\left ((1/4) \right )} \sin{\left (3 (1/4) \right )} + \sin{\left ((1/4) \right )} — 1$$

1 + sin(1/4) — cos(2/4) — sin(3/4)/2*sin(1/4)^2 + sin(1/4) — 1

$$-1 + \sin{\left (\frac{1}{4} \right )} + — \frac{1}{2} \sin^{2}{\left (\frac{1}{4} \right )} \sin{\left (\frac{3}{4} \right )} + — \cos{\left (\frac{2}{4} \right )} + \sin{\left (\frac{1}{4} \right )} + 1$$
Читайте также  sin(3*x)=cos(2*x)

-cos(1/2) + 2*sin(1/4) — sin(1/4)^2*sin(3/4)/2

$$- \cos{\left (\frac{1}{2} \right )} — \frac{1}{2} \sin^{2}{\left (\frac{1}{4} \right )} \sin{\left (\frac{3}{4} \right )} + 2 \sin{\left (\frac{1}{4} \right )}$$
Степени
$$- \frac{1}{2} \sin^{2}{\left (a \right )} \sin{\left (3 a \right )} + 2 \sin{\left (a \right )} — \cos{\left (2 a \right )}$$
Численный ответ

-cos(2*a) + 2*sin(a) — 0.5*sin(a)^2*sin(3*a)

Рациональный знаменатель
$$- \frac{1}{2} \sin^{2}{\left (a \right )} \sin{\left (3 a \right )} + 2 \sin{\left (a \right )} — \cos{\left (2 a \right )}$$
Объединение рациональных выражений
$$\frac{1}{2} \left(- \sin^{2}{\left (a \right )} \sin{\left (3 a \right )} + 4 \sin{\left (a \right )} — 2 \cos{\left (2 a \right )}\right)$$
Читайте также  x^2+9*x=0
Общее упрощение

2
sin (a)*sin(3*a)
-cos(2*a) + 2*sin(a) — —————-
2

$$- \frac{1}{2} \sin^{2}{\left (a \right )} \sin{\left (3 a \right )} + 2 \sin{\left (a \right )} — \cos{\left (2 a \right )}$$
Соберем выражение
$$- \frac{1}{2} \sin^{2}{\left (a \right )} \sin{\left (3 a \right )} + 2 \sin{\left (a \right )} — \cos{\left (2 a \right )}$$

sin(3*a) sin(5*a) 17*sin(a)
-cos(2*a) — ——— + ——— + ———
4 8 8

$$\frac{17}{8} \sin{\left (a \right )} — \frac{1}{4} \sin{\left (3 a \right )} + \frac{1}{8} \sin{\left (5 a \right )} — \cos{\left (2 a \right )}$$
Общий знаменатель

2
sin (a)*sin(3*a)
-cos(2*a) + 2*sin(a) — —————-
2

$$- \frac{1}{2} \sin^{2}{\left (a \right )} \sin{\left (3 a \right )} + 2 \sin{\left (a \right )} — \cos{\left (2 a \right )}$$
Читайте также  9/(x-5)=9/5
Комбинаторика

/ 2
— -4*sin(a) + 2*cos(2*a) + sin (a)*sin(3*a)/
———————————————
2

$$- \frac{1}{2} \left(\sin^{2}{\left (a \right )} \sin{\left (3 a \right )} — 4 \sin{\left (a \right )} + 2 \cos{\left (2 a \right )}\right)$$
Тригонометрическая часть

/ sin(a)*sin(3*a)
-1 + |2 + 2*sin(a) — —————|*sin(a)
2 /

$$\left(- \frac{1}{2} \sin{\left (a \right )} \sin{\left (3 a \right )} + 2 \sin{\left (a \right )} + 2\right) \sin{\left (a \right )} — 1$$
Раскрыть выражение

2 / 3 2
2 2 sin (a)* — sin (a) + 3*cos (a)*sin(a)/
sin (a) — cos (a) + 2*sin(a) — —————————————
2

$$- \frac{1}{2} \left(- \sin^{3}{\left (a \right )} + 3 \sin{\left (a \right )} \cos^{2}{\left (a \right )}\right) \sin^{2}{\left (a \right )} + \sin^{2}{\left (a \right )} + 2 \sin{\left (a \right )} — \cos^{2}{\left (a \right )}$$
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