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 = \binom{2m+1}{1} ext{cos}^{2m} \theta ext{sin } \theta + \binom{2... 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...
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Answer
Using De Moivre's theorem, we can expand the sine function in terms of cosine and sine. The stated equality holds by recognizing the binomial expansion of the sine function, yielding the given expression.
Step 2
Deduce that the polynomial...
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Answer
The polynomial can be deduced from the roots given by De Moivre's theorem. Specifically, the roots correspond to specific values determined by the cotangent function at prescribed intervals.
Step 3
Prove that...
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Answer
To prove this, we can utilize the symmetry of the cotangent function and properties of roots of polynomials. Summing these cotangent values gives us the desired relation as shown.
Step 4
You are given that...
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Answer
Given the cotangent relation, we can substitute into known summation series, leading to the derived inequality involving rac{ ext{π}^2}{6} and the series sum.
Step 5
By considering the triangle ABE, deduce that KL = a - x...
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Answer
In triangle ABE, the height from E to AB is expressed in terms of x. Therefore, KL can be deduced as the vertical distance from E down the triangle, yielding KL = a - x. The area of rectangle KLNM is thus given by 2a⋅(a−x).
Step 6
Find the volume of the tetrahedron ABCD.
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Answer
To find the volume of the tetrahedron ABCD, we can use the formula for the volume of a tetrahedron: Volume = \frac{1}{3} \times ext{Area of Base} \times ext{Height}. Here, the area of the base is 2a⋅(a−x), and the height is 2a, leading to the final volume expression.
