Question

2cos(3п-х)+sin(п+x)=-3​

Answer 1

cos(3π - x) ≡ cos(π - x) ≡ - cos(x),

sin(π + x) ≡ - sin(x),

исходное уравнение равносильно:

-2cos(x) - sin(x) = -3.

2cos(x) + sin(x) = 3.

-1 ≤cos(x) ≤ 1

-2 ≤ 2cos(x) ≤ 2

-1 ≤sin(x) ≤ 1

-3 ≤ 2cos(x) + sin(x) ≤ 3

значение 3 достигается лишь при условии:

cos(x) = 1 и sin(x) = 1, это противоречит основному тригонометрическому тождеству cos²x + sin²x ≡ 1

Ответ. Нет решений.

Answer 2

To solve the equation $2\cos(3\pi - x) + \sin(\pi + x) = -3$, we can use trigonometric identities and properties to simplify and solve for $x$.

First, let's rewrite the trigonometric functions using their properties:

$\cos(3\pi - x) = \cos(3\pi)\cos(x) + \sin(3\pi)\sin(x)$ $\cos(3\pi) = -1$ (because cosine is negative at $3\pi$) $\sin(3\pi) = 0$ (because sine is zero at $3\pi$)

So, we have:

$\cos(3\pi - x) = -\cos(x)$

Now, let's substitute this back into the equation:

$2(-\cos(x)) + \sin(\pi + x) = -3$

Now, let's work with $\sin(\pi + x)$. Using the angle sum identity for sine, we get:

$\sin(\pi + x) = \sin(\pi)\cos(x) + \cos(\pi)\sin(x)$

$\sin(\pi) = 0$ (because sine is zero at $\pi$) $\cos(\pi) = -1$ (because cosine is negative at $\pi$)

So, we have:

$\sin(\pi + x) = 0\cdot\cos(x) + (-1)\cdot\sin(x) = -\sin(x)$

Now, our equation becomes:

$2(-\cos(x)) -\sin(x) = -3$

Let's simplify it further:

$-2\cos(x) - \sin(x) = -3$

Now, we can add $\sin(x)$ to both sides of the equation to isolate $\cos(x)$:

$-2\cos(x) = -3 + \sin(x)$

Next, divide both sides by -2:

$\cos(x) = \frac{3}{2} - \frac{1}{2}\sin(x)$

Now, we can use the fact that $\sin^2(x) + \cos^2(x) = 1$ to rewrite $\sin(x)$ in terms of $\cos(x)$:

$\sin^2(x) = 1 - \cos^2(x)$

$\sin(x) = \sqrt{1 - \cos^2(x)}$

Now, substitute this into the equation:

$\cos(x) = \frac{3}{2} - \frac{1}{2}\sqrt{1 - \cos^2(x)}$

Now, you have an equation in terms of $\cos(x)$. You can solve for $\cos(x)$ and then find $x$ by taking the arccosine of the solution. Keep in mind that there may be multiple solutions, so you may need to check the intervals for which the equation is valid.