Question

Cos(-5x)=-0.5; ctg(-6x)=-1

Answer 1

$cos(-5x)= -0,5;\\cos5x=-0,5;\\5x= \pm \frac{2\pi }{3} +2\pi n, ~n\in\mathbb {Z} ;\\\\x= \pm\frac{2\pi }{15} +\frac{2\pi n }{5} , ~n\in\mathbb {Z}$

$ctg (-6x)=-1;\\-ctg6x=-1;\\ctg6x=1; \\6x= \frac{\pi }{4} +\pi n, ~n\in\mathbb {Z} ;\\x=\frac{\pi }{24} +\frac{\pi n }{6} , ~n\in\mathbb {Z}$

Answer 2

1) cos(-5x) = -0,5.

  Функция у = cosx - чётная функция, поэтому cos(-5x) = cos(5x).

   Поэтому получим уравнение:

   cos(5x) = -0,5,

   5х = ± arccos(-0,5) + 2πk, k ∈ Z,

   5х = ±(π - π/3) + 2πk, k ∈ Z,

   5х = ±2π/3 + 2πk , k ∈ Z,

   х = ±2π/15 + 2πk/5, k ∈ Z;

2) ctg(-6x) = -1.

    Функция у = ctgx - нечетная, поэтому ctg(-6x) = -ctg(6x).

    Поэтому получим уравнение

    -ctg(6x) = -1,

     ctg(6x) = 1,

     6х = π/4 + πn, n ∈ Z,

     х = π/24 + πn/6, n ∈ Z.

Answer 3

To solve the equations cos(-5x) = -0.5 and ctg(-6x) = -1, we can follow these steps:

1. Solve the equation cos(-5x) = -0.5 for values of x:

   Since we know that the cosine function is periodic, and the cosine of an angle is equal to -0.5 in the second and third quadrants, we can write:

   -5x = π + 2πn and -5x = 2π - π + 2πn, where n is an integer.

   Solve each equation for x:

   -5x = π + 2πn x = -(π + 2πn)/5

   -5x = 2π - π + 2πn x = (2π - π + 2πn)/-5 x = (π + 2πn)/5

2. Solve the equation ctg(-6x) = -1 for values of x:

   The cotangent function is the reciprocal of the tangent function. Since the tangent is negative in the fourth quadrant, we can write:

   -6x = π + πn, where n is an integer.

   Solve for x:

   x = -(π + πn)/6

So, the solutions to the equations are:

For the equation cos(-5x) = -0.5: x = -(π + 2πn)/5 x = (π + 2πn)/5

For the equation ctg(-6x) = -1: x = -(π + πn)/6

Where n is an integer. These are the general solutions to the given equations.