Let m be a positive integer - HSC - SSCE Mathematics Extension 2 - Question 8 - 2002 - Paper 1
Question 8
Let m be a positive integer.
(i) By using De Moivre's theorem, show that
$$ ext{sin}(2m+1) heta = \frac{(2m+1)}{1} \text{cos}^{2m}\theta \text{sin}(\theta) + \left... show full transcript
Worked Solution & Example Answer:Let m be a positive integer - HSC - SSCE Mathematics Extension 2 - Question 8 - 2002 - Paper 1
Step 1
By using De Moivre's theorem, show that sin(2m+1)θ = ...
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Answer
To demonstrate this, we use De Moivre's theorem which states that for any complex number in polar form, the nth power can be computed using trigonometric functions. Thus,
sin(nθ)=2i1(einθ−e−inθ)
Expanding both sides using the binomial theorem, we rearrange terms and apply known trigonometric identities to separate the coefficients, thereby arriving at the equation as required.
Step 2
Deduce that the polynomial p(x) = ...
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Answer
From the previously obtained sine expression, we can substitute x=sin(θ) and relate it to polynomial roots. The polynomial can be represented in terms of its roots, leading us to deduce that indeed it possesses distinct roots:
p(x)=(2m+1)xm−(2m+1)xm−1+...+(−1)m.
Step 3
Prove that cot(π/(2m+1)) + ... + cot(mπ/(2m+1)) = m(2m-1)/3.
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Answer
To prove this identity, we utilize the properties of cotangent functions as they relate to the angles defined by the polynomial's roots. Considering symmetry and the periodic nature of the cotangent, we can sum these cotangent terms as specified, leveraging trigonometric identities to reach the resultant formula.
Step 4
Deduce that π²/6 < ...
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Answer
Using the limit approach of cotθ and the relationships established by the previous steps, we can apply inequalities concerning the sum of reciprocals of squares. This can be framed through known convergence tests, yielding the required inequalities about 6π2.
Step 5
By considering the triangle ABE, deduce that KL = a - x, and find the area of the rectangle KLNM.
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Answer
In triangle ABE, with height AF = a and base AB = 2a, we recognize that the length KL is determined by the height diminishing with the distance x from midpoint P. Therefore, we conclude that:
KL=a−x.
The area of rectangle KLNM can then be calculated as:
Area=KL⋅(2a)=(a−x)(2a).
Step 6
Find the volume of the tetrahedron ABCD.
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Answer
To find the volume of tetrahedron ABCD, we apply the formula for the volume of a tetrahedron defined by base area and height:
V=31⋅Base Area⋅h,
where h is the perpendicular height from point D to the base formed by triangle ABE. Substituting values from the previously calculated areas gives us the full expression for volume.
