Question

Sin^4 (a) + cos^4 (a) + 1/2 sin^2 (2a)

Answer 1

Ответ:

sin^4 (a) + cos^4 (a) + 1/2 sin^2 (2a)=1

Объяснение:

Воспользуемся формулой синуса двойного угла (sin(2a)=2sin(a)cos(a))

sin^4(a)+cos^4(a)+2sin^2(a)*cos^2(a)

Уравнение представляет собой формулу квадрата суммы:

(a+b)^2=a^2+2ab+b^2

sin^4(a)+cos^4(a)+2sin^2(a)*cos^2(a) =(sin^2(a)+cos^2(a))^2

sin^2(a)+cos^2(a)=1  (Основное тригонометрическое тождество)

(sin^2(a)+cos^2(a))^2=1^2=1

Answer 2

To simplify the expression Sin^4(a) + Cos^4(a) + 1/2 Sin^2(2a), we'll use some trigonometric identities and algebraic manipulation.

1. Start with the double angle identity for sine: Sin(2a) = 2 * Sin(a) * Cos(a).
2. Square the double angle identity: Sin^2(2a) = (2 * Sin(a) * Cos(a))^2 = 4 * Sin^2(a) * Cos^2(a).

Now substitute this back into the original expression:

Sin^4(a) + Cos^4(a) + 1/2 * (4 * Sin^2(a) * Cos^2(a))

3. Distribute the 1/2 to both terms inside the parentheses:

Sin^4(a) + Cos^4(a) + 2 * Sin^2(a) * Cos^2(a)

4. Use the trigonometric identity: Sin^2(a) + Cos^2(a) = 1.

Now substitute this identity into the expression:

Sin^4(a) + (1 - Sin^2(a))^2 + 2 * Sin^2(a) * (1 - Sin^2(a))

5. Expand the square of the binomial:

Sin^4(a) + (1 - 2 * Sin^2(a) + Sin^4(a)) + 2 * Sin^2(a) - 2 * Sin^4(a)

6. Combine like terms:

2 * Sin^4(a) + 2 * Sin^2(a) - 2 * Sin^4(a)

7. Cancel out the common term:

2 * Sin^2(a)

So, the simplified expression is:

2 * Sin^2(a)