Mathematics - Trigonometric Equation Question with Solution | TestHub

(a) From the graphs of $y=|\tan 2x|$ and $y=\sin x$ on $[0,\pi ]$, there are 4 intersections (q). (b) $\tan 4A=1$ at $A=\frac{\pi }{16}$ gives $\frac{4t-4t^3}{1-6t^2+t^4}=1\Rightarrow 4t-4t^3+6t^2-t^4-1=0$, so the required value is 2 (s). (c) $\tan (p\cot x)=\tan \left(\frac{\pi }{2}-p\tan x\right)\Rightarrow p=\frac{n\pi +\frac{\pi }{2}}{\tan x+\cot x}=\frac{\pi }{2}\sin x\cos x(\mathrm{n}=0)$, so $P_{\max }=\frac{\pi }{4}$ and $\frac{4}{\pi }{P}_{\max }=1$ (p). (d) With $y={\cos }^{2}2x+2{\sin }^{2}x$, the other exponent is $3-y:5^y+5^{3-y}=126\Rightarrow 5^y=125$ or $1\Rightarrow y=3$ or 0. $y=3\Rightarrow \cos 2x=-1\Rightarrow x=\frac{\pi }{2},\frac{3\pi }{2}\Rightarrow \frac{2x}{\pi }=1,3;y=0$ is impossible (p, r).