Question

Существуют ли такие x, ,при которых sin5x+cos 5x=1,5 Ответ:нет, но НУЖНО РЕШИТЬ С ПОМОЩЬЮ ВВЕДЕНИЯ ВСПОМОГАТЕЛЬНОГО АРГУМЕНТА

Answer 1

sin5x+cos 5x=1,5  ;
√2*(1√2 * sin5x + 1√2 * cos 5x) =3/2 ;
√2*(sinπ/4 * sin5x +cosπ/4 * cos 5x) =3/2 ;
cos(5x - π/4) = 3/2√2  , но  3/(2√2) ≈3/(2*1,42) ≈1,061 >1, поэтому первоначальное  уравнение не имеет решения .

Answer 2

Can the equation sin(5x) + cos(5x) = 1.5 have solutions?

The equation sin(5x) + cos(5x) = 1.5 does not have any solutions. However, it can be solved using an auxiliary argument.

Explanation: To solve the equation sin(5x) + cos(5x) = 1.5, we can use the trigonometric identity sin^2(x) + cos^2(x) = 1. By rearranging the equation, we get:

sin(5x) + cos(5x) = 1.5 => sin(5x) + cos(5x) - 1.5 = 0

Now, let's introduce an auxiliary argument, let's say t, such that:

sin(5x) + cos(5x) - 1.5 = 0 => sin(5x) + cos(5x) - 1.5 = sin(t) + cos(t) - 1.5

By comparing the two equations, we can see that:

5x = t

Now, we need to find the values of t for which sin(t) + cos(t) - 1.5 = 0. Unfortunately, there are no values of t that satisfy this equation.

Therefore, the equation sin(5x) + cos(5x) = 1.5 does not have any solutions.