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Which of the following is equivalent to #sin9x#?

1) #sin8xsinx+cos8xcosx#
2) #4sin^3 3x-3sin3x#
3) #sin5xcos4x+cos4xsin5x#
4) #3sin3x-4sin^(3)3x#

Answer 1

Oliver Atkinson

Fourth option is equivalent to #sin9x#

#sin9x=3sin3x-4sin^(3)3x#

A reference is made to the triple angle identity for sine, which is

#sin3theta=3sintheta-4sin^3theta#.......(A)

Let us put #theta=3x#, then (A) becomes

#sin3(3x)=3sin(3x)-4sin^3(3x)#

or #sin9x=3sin3x-4sin^(3)3x#

Hence fourth option is equivalent to #sin9x#

First is #cos(8x-x)=cos7x#, second would be #sin(-9x)# and third is #2sin5xcos4x#.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

Which of the following is equivalent to #sin9x#?

1) #sin8xsinx+cos8xcosx# 2) #4sin^3 3x-3sin3x# 3) #sin5xcos4x+cos4xsin5x# 4) #3sin3x-4sin^(3)3x#

1) #sin8xsinx+cos8xcosx#
2) #4sin^3 3x-3sin3x#
3) #sin5xcos4x+cos4xsin5x#
4) #3sin3x-4sin^(3)3x#