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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 = cos⁵ x + 5i cos...

Answered by Tom R. • Further Mathematics tutor

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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

Answered by Cezar P. • Further Mathematics tutor

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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 int...

Answered by Adam T. • Further Mathematics tutor

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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=cos²(x)+2isin(x)cos(x)-sin²(x), which, according to de Moivre's cos²(x)+2is...

Answered by Cezar P. • Further Mathematics tutor

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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 c...

Answered by James K. • Further Mathematics tutor

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