Question

Если sin2A=0.5,то найдите sin^4+cos^4

Answer 1

${sin}^{4} \alpha + {cos}^{4} \alpha = {sin}^{4} \alpha + {cos}^{4} \alpha + 2 {sin}^{2} \alpha {cos}^{2} \alpha - 2 {sin}^{2} \alpha {cos}^{2} \alpha = {( {sin}^{2} \alpha + {cos}^{2} \alpha )}^{2} - 2 {sin}^{2} \alpha {cos}^{2} \alpha = 1 - 2 {sin}^{2} \alpha {cos}^{2} \alpha \\ \\ sin2 \alpha = 0.5 \\ 2sin \alpha cos \alpha = 0.5 \\ sin \alpha cos \alpha = \frac{1}{4} \\ {sin}^{2} \alpha {cos}^{2} \alpha = \frac{1}{16} \\ \\ 1 - 2 \times \frac{1}{16} = 1 - \frac{1}{8} = \frac{7}{8}$
Ответ: 7/8.

Answer 2

$\sin^4\alpha +\cos^4\alpha =(\sin^2\alpha)^2 +(\cos^2\alpha )^2=(\sin^2\alpha)^2+2\sin^2\alpha\cos^2\alpha+\\\\+(\cos^2\alpha )^2-2\sin^2\alpha\cos^2\alpha=(\sin^2\alpha+\cos^2\alpha)^2-\frac{1}{2} \cdot4\sin^2\alpha\cos^2\alpha=\\ \\ =1^2-\frac{1}{2} \cdot2^2\sin^2\alpha\cos^2\alpha=1-\frac{1}{2} \cdot(2\sin\alpha\cos\alpha )^2=1-\frac{1}{2} \cdot(\sin 2\alpha )^2=\\\\=1-\frac{1}{2} \cdot(0,5)^2=1-0,5\cdot0,25=1-0,125=0,875$

Ответ: $0,875$

Answer 3

To find the value of sin^4(A) + cos^4(A) given sin(2A) = 0.5, we can use trigonometric identities and algebraic manipulations.

Let's start with the given equation: sin(2A) = 0.5

Using the double-angle identity for sine, we have: sin(2A) = 2sin(A)cos(A)

So we can rewrite the given equation as: 2sin(A)cos(A) = 0.5

Now, let's focus on the expression we want to find: sin^4(A) + cos^4(A)

We know the trigonometric identity: sin^2(A) + cos^2(A) = 1

We can square this identity to get: (sin^2(A) + cos^2(A))^2 = 1

Expanding the left-hand side: (sin^2(A))^2 + 2sin^2(A)cos^2(A) + (cos^2(A))^2 = 1

Now, let's use another trigonometric identity: 2sin^2(A)cos^2(A) = sin^2(2A)

Substitute this into the previous equation: (sin^2(A))^2 + sin^2(2A) + (cos^2(A))^2 = 1

Now, we can rearrange this equation to get: (sin^2(A))^2 + (cos^2(A))^2 = 1 - sin^2(2A)

We know that sin^2(A) + cos^2(A) = 1, so (sin^2(A))^2 + (cos^2(A))^2 can also be written as: (sin^2(A))^2 + (1 - sin^2(A))^2 = 1 - sin^2(2A)

Now, expand and simplify: (sin^2(A))^2 + (1 - 2sin^2(A) + sin^2(A))^2 = 1 - sin^2(2A)

(sin^2(A))^2 + (1 - sin^2(A))^2 = 1 - sin^2(2A)

Now, recall that sin(2A) = 0.5: (sin^2(A))^2 + (1 - sin^2(A))^2 = 1 - (0.5)^2

(sin^2(A))^2 + (1 - sin^2(A))^2 = 1 - 0.25

(sin^2(A))^2 + (1 - sin^2(A))^2 = 0.75

Now, we can simplify further: (sin^2(A))^2 + (1 - sin^2(A))^2 = 3/4

We have a squared term and its complement squared, and we want to find sin^4(A) + cos^4(A), so let's rewrite the above equation using the following identities: 1 - sin^2(A) = cos^2(A)

(sin^2(A))^2 + cos^4(A) = 3/4

Now, let's solve for cos^4(A): cos^4(A) = 3/4 - (sin^2(A))^2

Since sin^2(A) + cos^2(A) = 1, we can rewrite sin^2(A) as: sin^2(A) = 1 - cos^2(A)

Now, substitute this into the previous equation: cos^4(A) = 3/4 - (1 - cos^2(A))^2

Now, let's expand and simplify: cos^4(A) = 3/4 - (1 - 2cos^2(A) + cos^4(A))

Now, we can solve for cos^4(A): cos^4(A) - cos^4(A) + 2cos^2(A) - 1 = 3/4

2cos^2(A) - 1 = 3/4

2cos^2(A) = 3/4 + 1

2cos^2(A) = 7/4

cos^2(A) = 7/4 * 1/2

cos^2(A) = 7/8

Now, we can find sin^2(A) using sin^2(A) = 1 - cos^2(A): sin^2(A) = 1 - 7/8

sin^2(A) = 1/8

Now, finally, we can find sin^4(A) + cos^4(A): sin^4(A) + cos^4(A) = (sin^2(A))^2 + (cos^2(A))^2

sin^4(A) + cos^4(A) = (1/8)^2 + (7/8)^2

sin^4(A) + cos^4(A) = 1/64 + 49/64

sin^4(A) + cos^4(A) = 50/64

sin^4(A) + cos^4(A) = 25/32

So, sin^4(A) + cos^4(A) is equal to 25/32.