Logs

Geometry Level 3

If log ⁡ cos ⁡ x sin ⁡ x ≥ 2 \log_{\cos x}\sin x\geq2 and x ∈ ( 0 , 3 π ) − n π 2 x\in(0,3\pi)-\frac{n\pi}{2} , where n n is any integer. Then sin ⁡ x \sin x lies in which of the following intervals?

[ 5 − 1 2 , 1 ] \left [\frac{\sqrt{5}-1}{2},1\right] ( 0 , 1 ) (0,1) [ 0 , 5 − 1 2 ] \left [0,\frac{\sqrt{5}-1}{2}\right] None of these

Shivam Jadhav by Shivam Jadhav , 22, India

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1 solution

Kushal Dey
Apr 14, 2020

For the equation to hold, both x should lie in (0,pi/2) so that both sinx and cosx are positive. However it is evident that ln(sinx) and ln(cosx) are always negative. Thus, ln(sinx)<=ln(1-sin^2(x)) And solving a quadratic equation you will find that sinx belongs to (0,(√5-1)/2).

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