Question

Sinx=1/2 x принадлежит [0;2п]

Answer 1

a)sinx=1/2

$x=\pi /6+2 \pi n$

И $x=5\pi /6+2\pi n$

b)Ответ $\pi /6$ и $5\pi /6$

Answer 2

To solve the equation sin(x) = (1/2)x over the interval [0, 2π], we can use numerical or graphical methods. However, we can also use algebraic methods to obtain an approximate solution.

One way to do this is to use the fact that sin(x) is approximately equal to x for small values of x. We can therefore start by assuming that x is small and solve the equation for this case. Then, we can use the solution as a starting point to refine our approximation iteratively.

Let's start by assuming that x is small and using the first two terms of the Taylor series for sin(x) and x:

sin(x) ≈ x - x^3/3!

We can then substitute this approximation into the original equation and solve for x:

x - x^3/3! = (1/2)x

Multiplying both sides by 3! and rearranging, we get:

6x - x^3 = 3π

This is a cubic equation, which can be solved using various methods. One common method is to use the rational root theorem to identify possible rational roots, and then use polynomial long division or synthetic division to find the remaining roots.

In this case, the rational root theorem suggests that any rational root of the equation must be a divisor of 3π. However, since 3π is irrational, the equation has no rational roots. Therefore, we need to use numerical or graphical methods to find an approximate solution.

One graphical method is to plot the two functions y = sin(x) and y = (1/2)x on the same graph and look for their intersection points. We can use a graphing calculator or software to do this easily. The intersection points correspond to the solutions of the equation.

Alternatively, we can use numerical methods such as Newton's method or the bisection method to find an approximate solution. These methods involve iterating a formula or algorithm that converges to a solution. The accuracy of the solution depends on the number of iterations and the initial guess.

In summary, the equation sin(x) = (1/2)x over the interval [0, 2π] can be solved approximately using algebraic, graphical, or numerical methods. The exact solution is irrational, but we can obtain an approximate solution with any desired degree of accuracy.