\( \begin{array}{ll}\text { 27. } w=\operatorname{sen} \alpha \cos \beta & \text { 28. } f(x, y)=x^{y} \\ \text { 29. } F(x, y)=\int_{y}^{x} \cos \left(e^{t}\right) d t & \text { 30. } F(\alpha, \beta)=\int_{\alpha}^{\beta} \sqrt{t^{3}+1} d t \\ \text { 31. } f(x, y, z)=x z-5 x^{2} y^{3} z^{4} & \text { 32. } f(x, y, z)=x \operatorname{sen}(y-z) \\ \text { 33. } w=\ln (x+2 y+3 z) & \text { 34. } w=z e^{x z} \\ \text { 35. } u=x y \operatorname{sen}^{-1}(y z) & \text { 36. } u=x^{\beta / z} \\ \text { 37. } h(x, y, z, t)=x^{2} y \cos (z / t) & \\ \text { 38. } \phi(x, y, z, t)=\frac{\alpha x+\beta y^{2}}{\gamma z+\delta y^{2}} \\ \text { 39. } u=\sqrt{x_{1}^{2}+x_{2}^{2}+\cdots+x_{n}^{2}} & \\ \text { 40. } u=\operatorname{sen}\left(x_{1}+2 x_{2}+\cdots+n x_{n}\right)\end{array} \)

Did you know that trigonometric functions like \( \sin \) and \( \cos \) trace their roots back to ancient civilizations? The Greeks used these concepts for astronomical calculations, leading to an explosion of mathematical knowledge. By the time of the Renaissance, these functions found their way into calculus and beyond, profoundly influencing fields including physics and engineering. So, the next time you calculate \( w = \sin \alpha \cos \beta \), remember you're carrying on a tradition that's thousands of years old! When it comes to real-world applications, consider how these mathematical expressions can model everything from wave motion to population dynamics. For instance, the function \( F(x, y) = \int_y^x \cos(e^t) dt \) can represent the way light waves interfere with one another, critical in optics. Understanding these concepts not only enhances educational pursuits, but can also lead to innovations in technology and science—who knows, perhaps the next big breakthrough in physics or engineering might spring from your calculations!