Question

Найти sin^4x-cos^4x, если -45° < x < 45° и cosx-sinx=(√0.72)

Answer 1

Ответ:-0,96

Объяснение:

Answer 2

To find the value of sin^4x - cos^4x, we first need to determine the values of sin(x) and cos(x) using the given equation cos(x) - sin(x) = √0.72.

We can start by squaring both sides of the given equation:

(cos(x) - sin(x))^2 = (√0.72)^2 cos^2(x) - 2*cos(x)*sin(x) + sin^2(x) = 0.72

Using the trigonometric identity cos^2(x) + sin^2(x) = 1, we can rewrite the equation as:

1 - 2*cos(x)*sin(x) = 0.72

Now, let's solve for cos(x)*sin(x):

2*cos(x)sin(x) = 1 - 0.72 2cos(x)*sin(x) = 0.28 cos(x)*sin(x) = 0.28 / 2 cos(x)*sin(x) = 0.14

Now, let's square both sides of the equation:

(cos(x)*sin(x))^2 = (0.14)^2 cos^2(x)*sin^2(x) = 0.0196

Next, we can use the Pythagorean identity sin^2(x) = 1 - cos^2(x) to express sin^2(x) in terms of cos^2(x):

sin^2(x) = 1 - cos^2(x)

Now, substitute sin^2(x) in the equation:

(1 - cos^2(x)) * cos^2(x) = 0.0196

Expand the equation:

cos^2(x) - cos^4(x) = 0.0196

Now, we need to find sin^4(x), and we can use the Pythagorean identity again to express sin^2(x) in terms of cos^2(x):

sin^2(x) = 1 - cos^2(x)

Substitute sin^2(x) in the equation:

sin^4(x) = (1 - cos^2(x))^2 sin^4(x) = (1 - cos^2(x)) * (1 - cos^2(x)) sin^4(x) = 1 - 2*cos^2(x) + cos^4(x)

Now, we can substitute the expression we found earlier for cos^2(x):

sin^4(x) = 1 - 2*(cos^2(x) - cos^4(x)) + cos^4(x) sin^4(x) = 1 - 2cos^2(x) + 2cos^4(x)

Finally, we can substitute the value of cos(x)*sin(x) we found earlier:

cos(x)*sin(x) = 0.14

sin^4(x) - cos^4(x) = 1 - 2cos^2(x) + 2cos^4(x) - cos^4(x)

sin^4(x) - cos^4(x) = 1 - 2*cos^2(x) + cos^4(x)

Now, we still need to find the value of cos^2(x). We can use the original equation cos(x) - sin(x) = √0.72 to get cos^2(x):

cos(x) - sin(x) = √0.72

Square both sides:

cos^2(x) - 2*cos(x)*sin(x) + sin^2(x) = 0.72

Since sin^2(x) = 1 - cos^2(x):

cos^2(x) - 2*cos(x)*sin(x) + (1 - cos^2(x)) = 0.72

Now, substitute cos(x)*sin(x) = 0.14:

cos^2(x) - 2*0.14 + (1 - cos^2(x)) = 0.72

Simplify the equation:

cos^2(x) - 0.28 + 1 - cos^2(x) = 0.72

Combine like terms:

1 - 0.28 = 0.72

0.72 = 0.72

Since the equation is true, the given value of cos(x) - sin(x) = √0.72 is valid.

Now, let's return to the expression for sin^4(x) - cos^4(x):

sin^4(x) - cos^4(x) = 1 - 2*cos^2(x) + cos^4(x)

Since we have already determined that cos(x) - sin(x) = √0.72, we can substitute cos^2(x) = 0.72 into the equation:

sin^4(x) - cos^4(x) = 1 - 2*0.72 + 0.72^2

Calculate the values:

sin^4(x) - cos^4(x) = 1 - 2*0.72 + 0.5184

sin^4(x) - cos^4(x) = 1 - 1.44 + 0.5184

sin^4(x) - cos^4(x) = 0.5184 - 0.44

sin^4(x) - cos^4(x) = 0.0784

Therefore, sin^4(x) - cos^4(x) = 0.0784 when -45° < x < 45° and cos(x) - sin(x) = √0.72.