given the identity sin(x+y)=sinx cosy + siny cosx
sin2x = 2 sinx cosx and
sin(2(x)+x) = sin 2x cos x + sinx cos 2x
using the last two identities gives
sin3x= 2 sinx cosx cosx + sinx cos2x
factoring the sinx we have
sin3x = sinx(2cosx cosx+cos2x)
which satisfies the requirement.
However, we can simplify further since cos 2x = cosx cosx - sinx sinx (a well known identity)
sin3x = sinx (2cosx cosx +cosx cosx - sinx sinx)
so sin3x= sinx(3cosx cosx - sinx sinx)
or sin 3x = 3.cos²x.sinx - sin³x
* * * * *
Good, but not good enough. The answer was required in terms of sin, not a mixture of sinx and cosx. Easily recitified, though, since cos²x = 1 - sin²x
Therefore sin3x = 3*(1-sin²x)*sinx - sin³x
= 3sinx - 4sin³x
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The derivative of y = sin(3x + 5) is 3cos(3x + 5) but only if x is measured in radians.
Yes, 3x and x are like terms because they have the same variable raised to the same power.
Multiply each term of the second expression by each of the terms in the first expression and add them together, collecting like terms together: (x - 6)(3x² + 10x - 1) = x(3x² + 10x - 1) + (-6)(3x² + 10x - 1) = (3x³ + 10x² - x) + (-18x² - 60x + 6) = 3x³ + 10x² - 18x² - x - 60x + 6 = 3x² + (10 - 18)x² + (-1 - 60)x + 6 = 3x² - 8x² - 61x + 6
The expression "negative x plus 3x" can be simplified by combining like terms. Negative x can be written as -1x, so the expression becomes -1x + 3x. When you combine -1x and 3x, you get 2x. Therefore, the simplified expression is 2x.
The expression "x minus 4x" can be simplified by combining like terms. When subtracting 4x from x, we are essentially subtracting 4 of the x terms from 1 of the x terms, resulting in a final answer of -3x. So, x minus 4x simplifies to -3x.