1) Cblevilos man \( 1 \sin (3 x)=-\operatorname{hin}^{\prime}(x)+\operatorname{sen}(x) \quad \left\lvert\, \quad \tan (3 x) \times \frac{1-3 \tan (x)}{\sin (x)-\tan (x)}\right. \) Montroc quil a) \( \cos ^{2}(A)+\cos ^{2}(B)+\cos ^{2}(C)=1-2 \cos (A) \cos (B) \cos (C) \) b) \( \sin (\lambda)+\sin (D)+\sin (C)=\operatorname{dos}\left(\frac{A}{2}\right) \cos \left(\frac{B}{2}\right) \cos \left(\frac{C}{2}\right) \) c) \( \cos (A)+\cos (D)+\cos (C)=1+4 \sin \left(\frac{A}{2}\right) \sin \left(\frac{B}{2}\right) \sin \left(\frac{C}{2}\right) \)

Did you know that the history of trigonometry dates back to ancient civilizations? The Babylonians and Egyptians had early forms of trigonometric concepts, using them for astronomy and architecture. The term "trigonometry" comes from the Greek words "trigonon" (triangle) and "metron" (measure). By the time of the Greeks, figures like Hipparchus and Ptolemy made significant advancements in understanding relationships between angles and sides, laying the groundwork for the trigonometry we use today! In real-world scenarios, trigonometry is not just reserved for math classrooms; it's essential in various fields! Architects use it to calculate structural load and angles, engineers rely on it for designing mechanical parts, and it's crucial in navigation for determining precise locations on maps. Additionally, video game developers often incorporate trigonometric calculations to create realistic movements and perspectives. So, the next time you're admiring a tall building or playing a video game, you can thank trigonometry!