1. **Stating the problem:** We are given several trigonometric identities and functions involving angle $\omega$ and variable $x$, and we need to analyze and solve the expressions and equations provided. 2. **Given identities and definitions:** - $\sin^2 \omega + \cos^2 \omega = 1$ (Pythagorean identity) - $\cos(x - \frac{\pi}{2}) = \sin x$ for all $x \in \mathbb{R}$ (co-function identity) - $f(x) = 2026 \sin 4x$ - $g(x) = -2026 \sin 4x$ - $\cos \omega = \frac{\sqrt{5}}{2}$ (note: $\frac{\sqrt{5}}{2} > 1$, which is impossible for cosine, so likely a typo or symbolic) - $f(x) = \sin(\pi - x) + \cos(\frac{\pi}{2} - x)$ - $f(x) = 2 \sin x$ - Domain: $0 \leq x \leq 2\pi$ - $\omega = A \hat{O} M$ (angle at origin between points A and M) 3. **Simplify $f(x) = \sin(\pi - x) + \cos(\frac{\pi}{2} - x)$:** Using identities: $$\sin(\pi - x) = \sin x$$ $$\cos\left(\frac{\pi}{2} - x\right) = \sin x$$ So, $$f(x) = \sin x + \sin x = 2 \sin x$$ This matches the given $f(x) = 2 \sin x$. 4. **Check $f(x) = 2026 \sin 4x$ and $g(x) = -2026 \sin 4x$:** These are two functions, $g(x) = -f(x)$ for the given $f(x)$. 5. **Solve equation $2f(x) - 3 = 0$ with $f(x) = 2 \sin x$:** $$2(2 \sin x) - 3 = 0$$ $$4 \sin x - 3 = 0$$ $$4 \sin x = 3$$ $$\sin x = \frac{3}{4}$$ 6. **Find $x$ in $[0, 2\pi]$ such that $\sin x = \frac{3}{4}$:** Using inverse sine: $$x = \arcsin\left(\frac{3}{4}\right)$$ Two solutions in $[0, 2\pi]$: $$x_1 = \arcsin\left(\frac{3}{4}\right)$$ $$x_2 = \pi - \arcsin\left(\frac{3}{4}\right)$$ 7. **Summary of solutions:** - $f(x) = 2 \sin x$ - $2f(x) - 3 = 0 \Rightarrow \sin x = \frac{3}{4}$ - Solutions for $x$ in $[0, 2\pi]$ are $x = \arcsin(\frac{3}{4})$ and $x = \pi - \arcsin(\frac{3}{4})$ 8. **Additional notes:** - The angle $\omega = A \hat{O} M$ is the angle at origin between points A and M on the unit circle. - The point M is approximately at $(-0.8, 0.6)$, which corresponds to $\cos \omega \approx -0.8$, $\sin \omega \approx 0.6$. - The given $\cos \omega = \frac{\sqrt{5}}{2}$ is not possible since cosine values are in $[-1,1]$. 9. **Desmos function for visualization:** We can plot $f(x) = 2 \sin x$ and $g(x) = -2026 \sin 4x$ to see their behavior.