Question

Sin(пи − x) − cos (пи/2+ x) = √3.

Answer 1

Ответ:

Пошаговое объяснение:

sin(π-x)=sinx

cos(π/2+x)=-sinx

sinx+sinx=√3

2sinx=√3

sinx=√3/2

x=(-1)ⁿπ/3+πn,n∈Z

Answer 2

To solve the equation $\sin(\pi - x) - \cos\left(\frac{\pi}{2} + x\right) = \sqrt{3}$, we can use trigonometric identities to simplify the expressions on the left-hand side of the equation and then solve for $x$.

1. Start with the given equation: $\sin(\pi - x) - \cos\left(\frac{\pi}{2} + x\right) = \sqrt{3}$

2. Use the angle sum identity for sine: $\sin(\pi - x) = \sin\pi \cos x - \cos\pi \sin x = 0 - (-1) \sin x = \sin x$

3. Use the angle sum identity for cosine: $\cos\left(\frac{\pi}{2} + x\right) = \cos\frac{\pi}{2} \cos x - \sin\frac{\pi}{2} \sin x = 0 \cdot \cos x - 1 \cdot \sin x = -\sin x$

Substitute these values back into the equation: $\sin x - (-\sin x) = \sqrt{3}$ $2 \sin x = \sqrt{3}$ $\sin x = \frac{\sqrt{3}}{2}$

4. The sine value of $\frac{\sqrt{3}}{2}$ corresponds to the angle $\frac{\pi}{3}$ in the unit circle.

5. However, we need to consider the domain of the original equation. The equation $\sin(\pi - x) - \cos\left(\frac{\pi}{2} + x\right) = \sqrt{3}$ doesn't have any constraints on the domain of $x$, so we need to find all possible solutions.

6. The solutions can be found by considering the periodicity of trigonometric functions. The sine function has a period of $2\pi$, so the angles $\frac{\pi}{3}$ and $\frac{\pi}{3} + 2\pi$ will both satisfy the equation.

So, the solutions for $x$ are: $x = \frac{\pi}{3} + 2\pi n$ where $n$ is an integer.

In summary, the solutions for the equation $\sin(\pi - x) - \cos\left(\frac{\pi}{2} + x\right) = \sqrt{3}$ are $x = \frac{\pi}{3} + 2\pi n$, where $n$ is an integer.