The first, third and fifth terms of an arithmetic progression are $$$2 \cos x,-6 \sqrt{3} \sin x$$$ and $$$10 \cos x$$$ respectively, where $$$\frac{1}{2} \pi< x< \pi$$$. Find the exact value of $$$x$$$. .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... ....................................................................................................................................................

Mathematics

IGCSE&ALevel

CAIE

Exam No：9709_w21_qp_12 Year：2021 Question No：5(a)

Answer:

$\left[\left(3^{\text {rd }}\right.\right.$ term $-1^{\text {st }}$ term $)=\left(5^{\text {th }}\right.$ term $-3^{\text {rd }}$ term $)$ leading to $\left.\ldots\right]$ $-6 \sqrt{3} \sin x-2 \cos x=10 \cos x+6 \sqrt{3} \sin x$ $[$ leading to $-12 \sqrt{3} \sin x=12 \cos x]$
OR
$\left[\left(1^{\text {st }}\right.\right.$ term $+5^{\text {th }}$ term $)=2 \times 3^{\text {rd }}$ term leading to...$] 12 \cos x=-12 \sqrt{3} \sin x$
Elimination of $\sin x$ and $\cos x$ to give an expression in $\tan x$
$\left[\tan x=-\frac{1}{\sqrt{3}}\right]$
$[x=] \frac{5 \pi}{6}$ only

Knowledge points:

1.5.4 use the identities

1.5.5 find all the solutions of simple trigonometrical equations lying in a specified interval (general forms of solution are not included).

1.6.3 use the formulae for the nth term and for the sum of the first n terms to solve problems involving arithmetic or geometric progressions (Including knowledge that numbers a,b,c are 'in arithmetic progression' if 2 b=a+c (or equivalent) and are 'in geometric progression' if (or equivalent) (Questions may involve more than one progression.)

Answer:

Knowledge points:

Solution:

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