Arithmetic progression with terms $3\cos x$, $-6\sin x$ and $9\cos x$

Key concept: This problem links arithmetic progression conditions with exact trigonometric values. The key is to use the equal difference between terms to form an equation in $x$, then use the AP sum formula once the first term and common difference are known.

Step by step

(a) Find the exact value of $x$

For an arithmetic progression, the middle term is the average of the first and third terms:

$2(-6\sin x)=3\cos x+9\cos x$

$-12\sin x=12\cos x$

$\sin x=-\cos x$

So:

$\tan x=-1$

Given that $\frac{\pi}{2}<x<\pi$, $x$ is in quadrant II, where sine is positive and cosine is negative. The angle satisfying $\tan x=-1$ in quadrant II is:

$\boxed{x=\frac{3\pi}{4}}$

(b) Find the exact sum of the first 25 terms

Substitute $x=\frac{3\pi}{4}$:

$\cos x=\cos\left(\frac{3\pi}{4}\right)=-\frac{\sqrt2}{2}$ $\sin x=\sin\left(\frac{3\pi}{4}\right)=\frac{\sqrt2}{2}$

So the first term is:

$a_1=3\cos x=-\frac{3\sqrt2}{2}$

The common difference is:

$d=a_2-a_1$

First find the second term using the AP rule:

$a_2=\frac{a_1+a_3}{2}=\frac{-\frac{3\sqrt2}{2}+9\left(-\frac{\sqrt2}{2}\right)}{2}=-3\sqrt2$

Then:

$d=-3\sqrt2-\left(-\frac{3\sqrt2}{2}\right)=-\frac{3\sqrt2}{2}$

Now use the sum formula:

$S_n=\frac{n}{2}\left(2a_1+(n-1)d\right)$

For $n=25$:

$S_{25}=\frac{25}{2}\left(2\left(-\frac{3\sqrt2}{2}\right)+24\left(-\frac{3\sqrt2}{2}\right)\right)$

$S_{25}=\frac{25}{2}\left(-3\sqrt2-36\sqrt2\right)$

$S_{25}=\frac{25}{2}(-39\sqrt2)$

$\boxed{S_{25}=-\frac{975\sqrt2}{2}}$

Pitfall alert

Do not try to solve the trigonometric equation by treating $\cos x$ and $\sin x$ as independent numbers. The AP condition gives a relationship between them. Also, keep the quadrant restriction in mind: $\tan x=-1$ has more than one solution, but only $\frac{3\pi}{4}$ lies in $\left(\frac{\pi}{2},\pi\right)$.

Try different conditions

If the question asked for the first $n$ terms instead of 25 terms, you would still find $a_1$ and $d$ first, then substitute into

$S_n=\frac{n}{2}\left(2a_1+(n-1)d\right).$

If the interval for $x$ were different, the equation $\tan x=-1$ could lead to a different exact angle.

Further reading

arithmetic progression, trigonometric identity, sum of series