Differentiation (Part-38) | S N De

 

$\,1(vi)\,\, 3\sin^2\theta+7\cos^2\theta=6 \\ \Rightarrow 3\tan^2\theta+7=6\sec^2\theta \\ [\text{Dividing both sides by }\,\,\cos^2\theta] \\ \Rightarrow 3\tan^2\theta+7=6(1+\tan^2\theta) \\ \Rightarrow 6+6\tan^2\theta-3\tan^2\theta-7=0 \\ \Rightarrow 3\tan^2\theta=1 \\ \Rightarrow \tan^2\theta=\frac 13 \\ \Rightarrow \tan\theta=\pm\frac{1}{\sqrt3} \\ \Rightarrow \tan\theta=\tan\left(\pm \pi/6\right) \\ \therefore \theta=n\pi \pm \pi/6,\,\,n \in \mathbb Z$ 

$\,1(vii)\,\,\tan p\theta=\cot q\theta \\ \Rightarrow \tan p\theta=\tan\left(\frac{\pi}{2}-q\theta\right) \\ \Rightarrow p\theta=n\pi+\left(\frac{\pi}{2}-q\theta\right) \\ \Rightarrow (p+q)\theta=\frac{(2n+1)\pi}{2} \\ \Rightarrow \theta=\frac{(2n+1)\pi}{2(p+q)},\,\,\, n\,\text{being any integer.}$

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$\,\,1(viii)\,\,\tan 5\theta+\cot2\theta=0 \\ \Rightarrow \frac{\sin5\theta}{\cos5\theta}+\frac{\cos2\theta}{\sin2\theta}=0 \\ \Rightarrow \frac{\sin5\theta\sin2\theta+\cos5\theta\cos2\theta}{\cos5\theta\sin2\theta}=0 \\ \Rightarrow \cos5\theta\cos2\theta+\sin5\theta\sin2\theta=0 \\ \Rightarrow \cos(5\theta-2\theta)=0 \\ \Rightarrow \cos3\theta=0 \\ \Rightarrow 3\theta=(2n+1)\frac{\pi}{2},\,\,n\,\,\,\text{being any integer.} \\ \Rightarrow \theta=(2n+1)\frac{\pi}{6}.$

$\,\,1(ix)\,\, \sin5\theta+\sin2\theta=0 \\ \Rightarrow 2\sin\frac{5\theta+2\theta}{2}\cos\frac{5\theta-2\theta}{2}=0 \\ \Rightarrow 2\sin\frac{7\theta}{2}\cos\frac{3\theta}{2}=0 \\ \therefore \sin\frac{7\theta}{2}=0\rightarrow(1), \\ \text{or,}\,\,\cos\frac{3\theta}{2}=0 \rightarrow(2).$

Hence, by $\,(1)\,$ we get, $\,\,\frac{7\theta}{2}=n\pi \Rightarrow \theta=\frac{2n\pi}{7}$

Again, by $\,(2)\,$ we get, $\,\,\frac{3\theta}{2}=(2n+1)\frac{\pi}2{} \\ \Rightarrow \theta=(2n+1)\pi/3,\,\,\,n \,\,\,\text{being any integer.}$

$\,\,1(x)\,\,\tan^2x=3\csc^2x-1 \\ \Rightarrow 1+\tan^2x=3\csc^2x \\ \Rightarrow \sec^2x=3\csc^2x\\ \Rightarrow 1/\cos^2x=3/\sin^2x \\ \Rightarrow \sin^2x=3\cos^2x \\ \Rightarrow \tan^2x=3 \\ \Rightarrow \tan x=\pm \sqrt3 \\ \Rightarrow \tan x=\tan\left(\pm \frac{\pi}{3}\right) \\ \Rightarrow x=n\pi \pm \frac{\pi}{3},\,\,\,n \,\,\,\text{being any integer.}$

$\,1(xi)\,\,\,\tan x+\cot x=2\\ \Rightarrow p+\frac 1p=2,\,\,\,\text{where,}\,\,p=\tan x \\ \Rightarrow p^2+1=2p \\ \Rightarrow p^2-2p+1=0 \\ \Rightarrow (p-1)^2=0 \\ \Rightarrow p=1 \\ \Rightarrow \tan x=1=\tan\frac{\pi}{4} \\ \Rightarrow x=n\pi+\pi/4,\,\,\,\text{where}\,\,\,n\in \mathbb Z$

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• COMPLEX NUMBERS
• QUADRATIC EQUATIONS
• TRIGONOMETRIC RATIOS OF MULTIPLE ANGLES
• RELATION AND MAPPING
• TRIGONOMETRIC RATIOS OF SUBMULTIPLE ANGLES
• TRIGONOMETRIC RATIOS OF COMPOUND ANGLES