Theory question 108
Trigonometry(a) Given that sin (A + B) = sin A cos B + cos A sin B, without using mathematical tables or calculator, evaluate sin 105°, leaving your answer in the surd form. (You may use 105° = 60° + 45°).
(b) The houses on the side of a particular street are assigned odd numbers, starting from 11. If the sum of the numbers is 551, how many houses are there?
(c) The 1st and 3rd term of a Geometric Progression (G.P.) are 2 and 2/9 respectively. Find: (i) the common difference, (ii) the 5th term
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(a) sin 105° = sin 60° cos 45° + cos 60° sin 45° = (√3/2)(√2/2) + (1/2)(√2/2) = (√6 + √2)/4.
(b) An A.P. with a = 11, d = 2: Sn = n/2[22 + 2(n − 1)] = n(n + 10) = 551.
n2 + 10n − 551 = 0, (n + 29)(n − 19) = 0, so n = 19 houses.
(c)(i) ar2 = 2/9 with a = 2, so r2 = 1/9 and the common ratio r = ±1/3.
(ii) T5 = ar4 = 2 × 1/81 = 2/81.
In a G.P. you look for a common ratio, not a common difference.