How do you write sin(4x) in terms of sin(x)? Thank you!

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How do you write sin(4x) in terms of sin(x)? Thank you!

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Answer 1 · 2017-09-24T16:45:39

$sin 4 x = 4 (sin x - 2 {sin}^{3} x) \sqrt{1 - {sin}^{2} x}$ $sin4x=4(sinx-2sin^3x)sqrt(1-sin^2x)$

Explanation:

$sin 4 x = 2 sin 2 x cos 2 x$ $sin4x=2sin2xcos2x$

= $2 \times 2 sin x cos x \times (1 - 2 {sin}^{2} x$ $2xx2sinxcosx xx(1-2sin^2x$

= $4 sin x cos x - 8 {sin}^{3} x cos x$ $4sinxcosx-8sin^3xcosx$

This is generally used expansion for $sin 4 x$ $sin4x$ , however to write in terms of pure $sin x$ $sinx$ , one can substitute $cos x$ $cosx$ with $\sqrt{1 - {sin}^{2} x}$ $sqrt(1-sin^2x)$ , although there could be issues regarding choosing appropriate sign based on which quadrant $x$ $x$ falls.

In such a case $sin 4 x = 4 (sin x - 2 {sin}^{3} x) \sqrt{1 - {sin}^{2} x}$ $sin4x=4(sinx-2sin^3x)sqrt(1-sin^2x)$

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How do you write sin(4x) in terms of sin(x)? Thank you!

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