Question

1. Sin^2x*tg^2x-cos^2x Sinx=1/32. Найти угол между лучами OА если OB начало координат A(-2; 2 корень из 3) B (5;5)

Answer 1

$cos {x}^{2} = 1 - sin {x}^{2} = \frac{ 8 }{9}$
$cosx = \frac{ \sqrt{8} }{3}$
$tgx = \frac{ \sin(x) }{ \cos(x) } = \frac{1}{ \sqrt{8} }$

Answer 2

I assume you have two separate questions. I'll answer them one by one:

1. To simplify the expression sin^2(x)*tan^2(x) - cos^2(x), we can use the identity tan^2(x) = sin^2(x)/cos^2(x). Substituting this in the expression, we get:

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

Now, we can factor out cos^2(x) from the first term in the expression:

(sin^2(x)*sin^2(x) - cos^4(x))/cos^2(x)

Using the identity sin^2(x) + cos^2(x) = 1, we can substitute sin^2(x) with 1 - cos^2(x):

((1 - cos^2(x))*sin^2(x) - cos^4(x))/cos^2(x)

Expanding the first term in the numerator, we get:

((sin^2(x) - cos^2(x)*sin^2(x)) - cos^4(x))/cos^2(x)

Simplifying further, we get:

(sin^2(x) - cos^2(x)*(1 - cos^2(x))) / cos^2(x)

Now, we can use the identity cos^2(x) = 1 - sin^2(x) to substitute cos^2(x) in the second term of the numerator:

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

Expanding the square in the second term of the numerator and simplifying, we get:

(2sin^4(x) - 2sin^2(x) + 1) / (1 - sin^2(x))

This expression cannot be simplified further without more information about x.

2. To find the angle between the rays OA and OB, we first need to find the direction vectors of these two rays. The direction vector of OA is the vector from the origin O to point A, which is:

OA = A - O = (-2, 2*sqrt(3))

Similarly, the direction vector of OB is the vector from the origin O to point B, which is:

OB = B - O = (5, 5)

To find the angle between these two vectors, we can use the dot product formula:

OA · OB = |OA| * |OB| * cos(theta)

where |OA| and |OB| are the magnitudes of the vectors OA and OB, and theta is the angle between them. Since the magnitudes of both vectors are sqrt(4 + 12) = 2*sqrt(3), we can simplify the equation to:

OA · OB = 12 * cos(theta)

To find the dot product, we can multiply the corresponding components of the two vectors and add them up:

OA · OB = (-2)(5) + (2sqrt(3))(5) = 10sqrt(3) - 10

Substituting this into the dot product equation, we get:

10*sqrt(3) - 10 = 12 * cos(theta)

Solving for cos(theta), we get:

cos(theta) = (10*sqrt(3) - 10) / 12

Using a calculator, we can find that cos(theta) ≈ 0.049, which means theta ≈ 87.5 degrees. Therefore, the angle between the rays OA and OB is approximately 87.5 degrees.