How do we derive the expansion for sin 3x?
- print Print
- list Cite
Expert Answers
calendarEducator since 2010
write12,544 answers
starTop subjects are Math, Science, and Business
I think by expansion you mean sin 3x in terms of sin x.
We start with the relations sin (x + y) = sin x* cos y + cos x*sin y and cos (x + y) = cos x* cos y – sin x*sin y
sin 3x = sin (2x + x) = sin 2x* cos x + cos...
(The entire section contains 144 words.)
Unlock This Answer Now
Start your 48-hour free trial to unlock this answer and thousands more. Enjoy eNotes ad-free and cancel anytime.
sin(2x+x) = sin2x*cosx + sin x*cos 2x
sin 2x = 2sinx*cos x
cos 2x = 1 - 2(sin x)^2
sin 3x = 2sin x*cos x*sin x + sin x*[1-2(sin x)^2]
sin 3x = 2(cos x)^2*sinx + sin x - 2 (sin x)^3
From the fundamental formula of trigonometry, we'll have:
(cos x)^2 = 1 - (sin x)^2
We'll try to express sin 3x in terms of sin x:
sin 3x = 2[1 - (sin x)^2]*sin x + sin x - 2(sin x)^3
sin 3x = 2sin x - 2(sin x)^3 + sin x - 2(sin x)^3
We'll combine like terms:
sin 3x = 3sin x - 4(sin x)^3
Student Answers