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Show, using de Moivre's theorem, that sin 5x = 16 sin^(5) x - 20 sin^(3) x + 5 sin x 

Using the question, first use de Moivre's theorem to say (cos x + i sin x)^5 = cos 5x + isin 5x. Now we can use a binomial expansion on the RHS to say cos 5x + isin 5x = cos 5 x + 5i cos 4 x sin x +10i 2 cos...
TR
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Expand (2x+3)^4

We will you use Pascal's triangle in order to find coefficients: 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 so, our coefficients will be 1,4,6,4,1 now, let's expand: (2x+3) 4 =(2x) 4 +4*(2x) 3 3+6 (2x) 2 3 2 +4 (2x)*3 3 ...
CP
4436 Views

Find the integrating factor of the following first order ODE: dx/dt = -2tx +t.

Firstly rearrange the differential equation to fit the form dx/dt +P(t)x = Q(t). The integrating factor is then found by using the formula: u = EXP(INTG(P(t))). We know that P(t) = 2t and so by integrating w...
AT
8575 Views

Using your knowledge of complex numbers, such as De Moivre's and Euler's formulae, verify the trigonometric identities for the double angle.

de Moivre's: (cos(x)+isin(x)) n =cos(nx)+isin(nx) set n=2 (cos(x)+isin(x)) 2 =cos 2 (x)+2isin(x)cos(x)-sin 2 (x), which, according to de Moivre's cos 2 (x)+2isin(x)cos(x)-sin 2 (x)=cos(2x)+isin(2x) We notice...
CP
4992 Views

A particle is launched from the top of a cliff of height 87.5m at time t=0 with initial velocity 14m/s at 30 deg above the horizontal, Calculate: a) maximum height reached above bottom of cliff; b)horizontal distance travelled before hitting the ground.

For parabolic motion: vertical and horizontal components can be treated independently. x represents horizontal motion, which obeys constant velocity equations, y denotes vertical motion, which obeys the cons...
JK
4163 Views