Решение тригонометрического уравнения

Breaking down the equation

I’m using trigonometric identities to simplify the equation. Substituting $\sin(2x)$ and $\cos(2x)$, the equation becomes $2 \sin x \cos x + (\cos^2 x - \sin^2 x) + \sin x + \cos x + 1$.

Simplifying with identities

I’m using angle addition formulas to rewrite $\sin(2x) + \cos(2x)$ as $\sqrt{2}\sin(2x + \pi/4)$ and $\sin(x) + \cos(x)$ as $\sqrt{2}\sin(x + \pi/4)$.

Transforming the equation

I’m using trigonometric identities to transform the equation into $2\sqrt{2} \sin\left(\frac{3x}{2} + \frac{\pi}{4}\right) \cos\left(\frac{x}{2}\right) = -1$.

Rewriting and simplifying

I'm reworking the expression by defining $u = \sin x + \cos x$ and leveraging identity transformations to express the equation in terms of $\sin x + \cos x$.

Formulating a new strategy

I’m mapping out strategies to express $\sin(2x) + \cos(2x)$ in terms of $\sin x + \cos x$, such as defining new variables $u$ and $v$.

Combining terms

I’m consolidating $\sin(2x + \pi/4)$ and $\sin(x + \pi/4)$ using trigonometric identities, leading to $\sin\left(\frac{3x}{2} + \frac{\pi}{4}\right) \cos\left(\frac{x}{2}\right) = -\frac{1}{2\sqrt{2}}$.

Ensuring consistency

I’m mapping out the equation with $\sin\left(\frac{3x}{2} + \frac{\pi}{4}\right) \cos\left(\frac{x}{2}\right)$ to verify it aligns with the original equation.

Refining the approach

I’m considering simplifying the equation by substituting $u = \sin x + \cos x$. This method will help express everything in terms of $u$.

Piecing together

I’m pulling together expressions for $2 \sin x \cos x$ and $\cos^2 x - \sin^2 x$ using $u$ and $v$ to find common terms and simplify the equation further.

Reworking identities

I’m re-examining the identity $\cos^2 x - \sin^2 x = \cos(2x)$ and expressing it through $u$ and $v$ to find common terms for further simplification.

Translating the equation

I’m translating the equation into terms of $u$ and $v$ to express $\sin(2x) + \cos(2x)$ as $(u^2 - 1) + uv$, aiming for further simplification.

Piecing it together

I’ve figured out how to express $u^2 + uv + u = 0$ and found $v = \pm \sqrt{2 - u^2}$.

Evaluating solutions

First, I’m setting $u = 0$ and seeing if $\sin x + \cos x = 0$ works. Then, I’m checking that against the original equation to confirm its validity.

Mapping out solutions

I’m figuring out two solution sets: $x = -\frac{\pi}{4} + n\pi$ and $\cos x = -\frac{1}{2}$ leading to $x = \pm \frac{2\pi}{3} + 2k\pi$.

Examining solution constraints

I’m verifying the solutions are consistent with the original equation, ensuring no constraints were overlooked, particularly for $\cos x = -\frac{1}{2}$.

Verifying consistency

I’m making sure that $\cos x = -\frac{1}{2}$ and corresponding $\sin x$ values fit with the original equation, checking solutions like $x = \frac{2\pi}{3} + 2k\pi$.

Ensuring solution clarity

I’m refining the solution set to $x = -\frac{\pi}{4} + n\pi$ and $x = \frac{2\pi}{3} + 2n\pi$ or $x = \frac{4\pi}{3} + 2n\pi$ for clarity.

Thought about solving trigonometric equation for 1m 23s