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Use De Moivre's Theorem to show that if z = cos(q)+isin(q), then (z^n)+(z^-n) = 2cos(nq) and (z^n)-(z^-n)=2isin(nq).

De Moivre's Theorem states that if z = cos(q)+isin(q), then z n = (cos(q)+isin(q)) n = cos(nq)+isin(nq) But then z -n = cos(-nq)+isin(-nq). Now, cos(-p)=cos(p), as cosine is a symmetric (even) function, and ...
DA
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Two lines have equations r = (1,4,1)+s(-1,2,2) and r = (2,8,2)+t(1,3,5). Show that these lines are skew.

Recall that for two lines to be skew they must satisfy two conditions: 1) They must not be parallel. 2) They must not intersect. We shall check each condition individually. Condition 1 The general vector equ...
DA
Answered by Dorian A. • Maths tutor
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Prove that matrix multiplication is not commutative.

At GCSE level, proof questions are relatively rare and largely will all require a similar sort of approach. The difference with A Level is that the syllabus contains more than one method of proof. You will b...
SF
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A satellite is in a stationary orbit above a planet of mass 8.9 x 10^25 kg and period of rotation 1.2 x 10^5 s. Calculate the radius of the satellite's orbit from the centre of the planet.

A body in a stationary orbit will always remain above the same point on the planet as it orbits. For a body to be in such an orbit, it must rotate around the planet in the same direction as the spin of the p...
DA
Answered by Dorian A. • Physics tutor
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How do you find the minimum of the equation sin^2(x) + 4sin(x)?

The first step to solving this problem is to treat it as a normal quadratic equation; if you are struggling with comparing our equation to a normal quadratic, try substituting sin(x) = y into the equation as...
KS
Answered by Kirsten S. • Maths tutor
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