Bạn đang xem: Trigonometry
$$sin(4x) = 4 sin(x) cos(x) cos(2x)$$
The book does some magic and gets $$2(2sin(x)cos(x))cos(2x)$$
This makes no sense to me, if I expand that I get $$4sin(x)cos(2x)cos(2x)$$ which is not equal.
Everything starts with $$sin(a+b)=sin acos b+cos asin b$$ This is an identity, it holds for all $a$ and $b$. In particular, you"re allowed to replace $b$ with $a$, so long as you do it consistently throughout, and you get $$sin2a=2sin acos a$$ Stop me if you didn"t follow this. Now we can replace $a$ everywhere with $2x$ and get $$sin 4x=2sin2xcos2x$$ Now there"s a $sin2x$ in that formula; we can use double-angle on it to get $$sin4x=2(2sin xcos x)cos2x$$ Now multiplication is associative, which means as long as all we"re doing is multiplication, we don"t need parentheses. On the right side, we"re multiplying 5 things: $$sin4x=2 imes2 imessin x imescos x imescos2x$$ Finally, $2 imes2=4$, so $$sin4x=4sin xcos xcos2x$$
Thanks for contributing an answer to fundacionfernandovillalon.comematics Stack Exchange!Please be sure to answer the question. Provide details and share your research!
But avoid …Asking for help, clarification, or responding to other answers.Making statements based on opinion; back them up with references or personal experience.
Use fundacionfernandovillalon.comJax to format equations. fundacionfernandovillalon.comJax reference.
Xem thêm: Phân Tích Tâm Trạng Thúy Kiều Trong Đoạn Trích Trao Duyên, Top 16 Bài
To learn more, see our tips on writing great answers.
Post Your Answer Discard
Not the answer you're looking for? Browse other questions tagged trigonometry or ask your own question.
How does $cos x cos(y − x) − sin y sin(x − y)$ reduce to $cos x$ as a function of one variable?
Finding $sin 2x$ from transforming $sin^4 x+ cos^4 x = frac79$ using trigonometric identities
Site design / logo © 2022 Stack Exchange Inc; user contributions licensed under cc by-sa. Rev2022.4.14.41981