Question

Sin^2 (x) - 7 sin (x) cos (x)= -2

Answer 1

Ответ:

Объяснение:

Sin² (x) - 7 sin (x) cos (x)+2(Sin² (x)+cos (x))=0

Sin² (x) - 7 sin (x) cos (x)+2Sin² (x)+2cos² (x)=0    / cos² (x)

tg² X-7tg X +2tg²X+2=0

3tg² X-7tg X +2=0    tg² X=к

3к² -7к +2=0

к=(7±√(49-4*3*2))/(2*3)

к=(7±√(49-24))/6

к=(7±5)/6

к₁=2/6=1/3   tg² X=1/3       tg X =±√3/3

к₂=12/6=2     tg² X=2         tg X=±√2

tg X₁ =-√3/3               X₁ =arctg(-√3/3)        X₁ = 5п/6 +пк

tg X₂ =+√3/3               X₂ =arctg(+√3/3)        X₂ =п/6 +пк

tg X₃ =-√2                     X ₃=arctg(-√2)       Х₃≈ 0.6959п+пк

tg X ₄=+√2                  X ₄=arctg(+√2)          Х₄≈0,304п+пк

Answer 2

To solve the equation sin^2(x) - 7sin(x)cos(x) = -2, we can use some trigonometric identities to simplify it and then solve for sin(x) and cos(x).

Start by using the double angle identity for sine:

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

Now, let's rewrite the equation:

sin^2(x) - 7sin(x)cos(x) = -2

sin^2(x) - 7/2 * 2sin(x)cos(x) = -2

sin^2(x) - 7/2 * sin(2x) = -2

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

sin^2(x) - 7/2 * [2sin(x)cos(x)] = -2

sin^2(x) - 7/2 * [sin(2x)] = -2

Now, substitute the double angle identity back in:

sin^2(x) - 7/2 * sin^2(x) = -2

Now, combine the terms with sin^2(x):

(1 - 7/2) * sin^2(x) = -2

-5/2 * sin^2(x) = -2

Now, isolate sin^2(x):

sin^2(x) = (-2) / (-5/2)

sin^2(x) = (2) / (5/2)

sin^2(x) = 2 * (2/5)

sin^2(x) = 4/5

Now, take the square root of both sides to find sin(x):

sin(x) = ±√(4/5)

sin(x) = ±2/√5

So, sin(x) is either sin(x) = 2/√5 or sin(x) = -2/√5.

Now, let's find cos(x) using the Pythagorean identity for sine:

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

cos^2(x) = 1 - (4/5)

cos^2(x) = 1/5

cos(x) = ±√(1/5)

cos(x) = ±1/√5

So, cos(x) is either cos(x) = 1/√5 or cos(x) = -1/√5.

Therefore, the solutions for the equation sin^2(x) - 7sin(x)cos(x) = -2 are:

1. sin(x) = 2/√5 and cos(x) = 1/√5
2. sin(x) = -2/√5 and cos(x) = -1/√5