Trigonometry! #66

Geometry Level 3

Find the values of $x$ in $(-\pi,\pi)$ which satisfy the equation. $8^{(1+\lvert\cos x\rvert+\cos^{2}x+\lvert\cos^{3}x\rvert+...)}=64$

Express the four values of $x$ in the form of $±\frac {a\pi}{b}$ and $±\frac {c\pi}{d}$ such that $a,b,c$ and $d$ are integers and where $gcd(a,b)=gcd(c,d)=1$ . Enter the value of $a+b+c+d$ .

This problem is part of the set Trigonometry .

The answer is 9.

1 solution

Omkar Kulkarni
Feb 21, 2015

$\lvert1\rvert+\lvert\cos x\rvert+\lvert\cos^{2}x\rvert+\lvert\cos^{3}x\rvert+\cdots$ forms an infinite GP, which sums to $\frac{1}{1-\lvert\cos x\rvert}$ . Hence we have $8^{\frac{1}{1-\lvert\cos x\rvert}}=8^{2} \\ \frac{1}{1-\lvert\cos x\rvert} = 2 \\ \lvert\cos x\rvert=\frac{1}{2} \\ \cos x = \pm\frac{1}{2} \\ \boxed{x=\pm\frac{\pi}{3},\pm\frac{2\pi}{3}}$

Trigonometry! #66

The answer is 9.

1 solution

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