How do you change $2 sin 7 x cos 4 x$ $2 \sin 7x \cos 4x$ into a sum?

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How do you change $2 sin 7 x cos 4 x$ $2 \sin 7x \cos 4x$ into a sum?

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Answer 1 · 2018-05-23T22:50:38

sin 11x + sin 3x

Explanation:

$f (x) = 2 cos 4 x . sin 7 x$ $f(x) = 2cos 4x. sin 7x$
Note that $sin 7 x = cos (\frac{π}{2} - 7 x)$ $sin 7x = cos (pi/2 - 7x)$ (complementary arcs)
$f (x) = 2 cos 4 x . cos (\frac{π}{2} - 7 x)$ $f(x) = 2 cos 4x.cos (pi/2 - 7x)$
Reminder of trig identity:
$2 cos a . cos b = cos (a - b) + cos (a + b)$ $2cos a.cos b = cos (a - b) + cos (a + b)$
In this case:
$a - b = 4 x - (\frac{π}{2} - 7 x) = 11 x - \frac{π}{2}$ $a - b = 4x - (pi/2 - 7x) = 11x - pi/2$
$a + b = 4 x + \frac{π}{2} - 7 x = \frac{π}{2} - 3 x$ $a + b = 4x + pi/2 - 7x = pi/2 - 3x$
$f (x) = cos (11 x - π 2) + cos (\frac{π}{2} - 3 x) = cos (\frac{π}{2} - 11 x) + sin 3 x$ $f(x) = cos (11x - pi2) + cos (pi/2 - 3x) = cos (pi/2 - 11x) + sin 3x$
$f (x) = sin 11 x + sin 3 x$ $f(x) = sin 11x + sin 3x$ (complementary arcs)

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How do you change $2 sin 7 x cos 4 x$ $2 \sin 7x \cos 4x$ into a sum?

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