Question

Sinx-cosx=3/корень из 10 найти cos4x

Answer 1

(sin x-cos x)^2 = 9/10
sin^2 x - 2sin x*cos x + cos^2 x = 9/10
1 - sin 2x = 9/10
sin 2x = 1/10
Cos 4x = 1 - 2sin^2 (2x) = 1 - 2/100 = 98/100 = 0,98

Answer 2

To find the value of cos(4x), we can use the identity for cos(2x):

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

Let's first solve the given equation:

sin(x) - cos(x) = 3/√10.

Now, let's square both sides of this equation to eliminate the square root:

(sin(x) - cos(x))^2 = (3/√10)^2.

(sin^2(x) - 2sin(x)cos(x) + cos^2(x)) = 9/10.

Now, we know that sin^2(x) + cos^2(x) = 1 (trigonometric identity), so we can substitute it in:

(1 - 2sin(x)cos(x)) = 9/10.

Now, we can isolate 2sin(x)cos(x):

2sin(x)cos(x) = 1 - 9/10.

2sin(x)cos(x) = 1/10.

sin(2x) = 1/10.

Now, we can use the double angle identity for sine:

sin(2x) = 2sin(x)cos(x).

Substitute the value of sin(2x):

2sin(x)cos(x) = 1/10.

Now, let's go back to the identity for cos(2x):

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

Substitute the value of sin^2(x):

cos(2x) = 1 - 2 * (1/10)^2.

cos(2x) = 1 - 2/100.

cos(2x) = 1 - 0.02.

cos(2x) = 0.98.

Now, we are almost there. We want to find cos(4x), so we'll use the double angle identity for cosine:

cos(4x) = 2cos^2(2x) - 1.

Substitute the value of cos(2x):

cos(4x) = 2 * (0.98)^2 - 1.

cos(4x) = 2 * 0.9604 - 1.

cos(4x) = 1.9208 - 1.

cos(4x) = 0.9208.

Therefore, the value of cos(4x) is approximately 0.9208.