Question

Sin³A+cos³A если sinA+cosA=-0,8

Answer 1

Надеюсь, всё понятно...

Answer 2

To find the value of sin³A + cos³A, we first need to determine the value of sinA and cosA. We are given that sinA + cosA = -0.8.

Let's use a trigonometric identity to find sin³A + cos³A:

sin³A + cos³A = (sinA)³ + (cosA)³ - 3sinAcosA(sinA + cosA)

We have sinA + cosA = -0.8, so we can rewrite the above equation as:

sin³A + cos³A = (sinA)³ + (cosA)³ - 3sinAcosA(-0.8)

Now, we need to find the values of (sinA)³ and (cosA)³.

Since sinA + cosA = -0.8, we can use the identity: (sinA + cosA)² = sin²A + 2sinAcosA + cos²A = 1

Squaring the given equation:

(-0.8)² = sin²A + 2sinAcosA + cos²A 0.64 = sin²A + 2sinAcosA + cos²A

Now, we know that sin²A + cos²A = 1 (from the identity sin²A + cos²A = 1), so we can rewrite the equation:

0.64 = 1 + 2sinAcosA

Now, we can find sinAcosA:

2sinAcosA = 0.64 - 1 2sinAcosA = -0.36

Next, we need to find (sinA)² and (cosA)²:

(sinA)² + (cosA)² = 1

Since we know (sinA)³ and (cosA)³ appear in the original expression, we can cube the equation above:

(sinA)⁶ + (cosA)⁶ + 2(sinA)³(cosA)³ = 1

Now, we need to find (sinA)⁶ and (cosA)⁶. Let's square the equation sinA + cosA = -0.8 again:

(sinA + cosA)² = (-0.8)² sin²A + 2sinAcosA + cos²A = 0.64

Using the identity sin²A + cos²A = 1:

1 + 2sinAcosA = 0.64

Now, let's square this equation once more:

(1 + 2sinAcosA)² = (0.64)² 1 + 4sinAcosA + 4(sinAcosA)² = 0.4096

We know that 2sinAcosA = -0.36, so let's substitute:

1 - 4(0.36) + 4(sinAcosA)² = 0.4096 1 - 1.44 + 4(sinAcosA)² = 0.4096

Now, let's solve for (sinAcosA)²:

4(sinAcosA)² = 0.4096 - 1 + 1.44 4(sinAcosA)² = 0.8496

(sinAcosA)² = 0.8496 / 4 (sinAcosA)² = 0.2124

Now, we can find sinAcosA:

sinAcosA = ±√0.2124

Since sinA + cosA = -0.8, both sinA and cosA must be negative. Therefore, sinAcosA = -√0.2124.

Now, we can find sin³A + cos³A using the previously derived expression:

sin³A + cos³A = (sinA)³ + (cosA)³ - 3sinAcosA(sinA + cosA)

sin³A + cos³A = (sinA)³ + (cosA)³ - 3sinAcosA(-0.8)

sin³A + cos³A = (sinA)³ + (cosA)³ + 2.4sinAcosA

sin³A + cos³A = (sinA)³ + (cosA)³ + 2.4(-√0.2124)

Now, we need to find the individual values of (sinA)³ and (cosA)³. Let's use the following identity:

(sinA + cosA)³ = sin³A + cos³A + 3sinAcosA(sinA + cosA)

We know that (sinA + cosA)³ = (-0.8)³ = -0.512. Now we can find (sinA)³ + (cosA)³:

(sinA)³ + (cosA)³ = (-0.512 - 3sinAcosA(sinA + cosA)) / -2.4

(sinA)³ + (cosA)³ = (-0.512 - 3(-√0.2124)(-0.8)) / -2.4

(sinA)³ + (cosA)³ = (-0.512 - 3(-√0.2124)(-0.8)) / -2.4

(sinA)³ + (cosA)³ ≈ -0.299

Finally, we can find sin³A + cos³A:

sin³A + cos³A ≈ -0.299 + 2.4(-√0.2124)

sin³A + cos³A ≈ -0.299 - 2.4√0.2124

sin³A + cos³A ≈ -0.299 - 1.472

sin³A + cos³A ≈ -1.771

So, sin³A + cos³A ≈ -1.771.