Question

А)1/cos^2x+1/sin(3п/2+x)=6 решить уравнение б)[-7п/2;-2п] найти какие точки входят в этот промежуток

Answer 1

$\frac{1}{cos^2x}-\frac{1}{cosx}=6, cosx\neq 0$

$6cos^2x+cosx-1=0$

$cosx=\frac{1}{3}, x=arccos(1/3)+2\pi k; x=\pi -arccos(1/3)+2\pi k$, где k - целое

$cosx=-\frac{1}{2}, x=\frac{2\pi}{3}+2\pi n; x=\frac{4\pi}{3}+2\pi n$, где n - целое

б) Дорешаете сами, образец во вложении

Answer 2

А) To solve the equation 1/cos^2(x) + 1/sin(3π/2 + x) = 6, we need to find the values of x that satisfy the equation. Let's start by simplifying the left-hand side of the equation:

1/cos^2(x) + 1/sin(3π/2 + x) = 1/cos^2(x) + 1/sin(π/2 - x)

Now, we know that sin(π/2 - x) = cos(x). Substituting this into the equation:

1/cos^2(x) + 1/cos(x) = 6

To solve this equation, let's find a common denominator for the two terms on the left-hand side:

(1 + cos(x)) / cos^2(x) = 6

Now, cross-multiply:

1 + cos(x) = 6 * cos^2(x)

Rearrange the equation to a quadratic form:

6 * cos^2(x) - cos(x) - 1 = 0

Now, let's solve this quadratic equation. We can use the quadratic formula:

For a quadratic equation of the form ax^2 + bx + c = 0, the solutions for x are given by:

x = (-b ± √(b^2 - 4ac)) / 2a

In our case, a = 6, b = -1, and c = -1. Let's calculate the discriminant:

Discriminant (D) = b^2 - 4ac = (-1)^2 - 4 * 6 * (-1) = 1 + 24 = 25

Since the discriminant is positive, we have two distinct real solutions for x:

x = (1 + √25) / 2 * 6 x = (1 + 5) / 12 x = 6/12 x = 1/2

x = (1 - √25) / 2 * 6 x = (1 - 5) / 12 x = -4/12 x = -1/3

So, the solutions to the equation are x = 1/2 and x = -1/3.

б) To find which points are in the interval [-7π/2, -2π], we need to determine all the values of x that lie within this range. The interval notation [-7π/2, -2π] means all x-values between -7π/2 and -2π, including both endpoints.

In this interval, -7π/2 ≤ x ≤ -2π.

So, any value of x that satisfies this inequality is within the specified interval. For example, x = -3π would be within the interval since it satisfies the inequality:

-7π/2 ≤ -3π ≤ -2π

However, x = -8π would not be within the interval since it does not satisfy the inequality:

-7π/2 ≤ -8π (false)

So, any value of x between -7π/2 and -2π, inclusive, is in the interval [-7π/2, -2π].