If 3costheta=5sintheta, then the value of frac{5sintheta-2sec^3theta+2costheta}{5sintheta+2sec^3theta-2costheta} is:

Q: If $3\cos\theta=5\sin\theta,$ then the value of $\frac{5\sin\theta-2\sec^3\theta+2\cos\theta}{5\sin\theta+2\sec^3\theta-2\cos\theta}$ is:

We have, $3 \cos\theta=5\sin\theta$ $\frac{\cos\theta}{\sin\theta}=\frac{5}{3 }$ ${\cot\theta}=\frac{5}{3} $ In $\triangle\text{ABC}, $ $\text{AC}^2=\text{AB}^2+\text{BC}^2$ $\Rightarrow \text{AC}^2=(3)^2+(5)^2$ $\Rightarrow \text{AC}^2=9+25$ $\Rightarrow \text{AC}^2=34$ $\Rightarrow \text{AC}=\sqrt{34}$ $\therefore \sin\theta=\frac{3}{\sqrt{34}}\ \cos\theta=\frac{5}{\sqrt{34}}\ \sec\theta=\frac{\sqrt{34}}{{5}}$ Now, $\frac{5\sin\theta-2\sec^3+2\cos}{5\sin\theta+2\sec^3-2\cos}$ $=\frac{5\times\frac{3}{\sqrt{34}}-2\Big(\frac{\sqrt{34}}{5}\Big)^3+2\times\frac{5}{\sqrt{34}}}{5\times\frac{3}{\sqrt{34}}+2\Big(\frac{\sqrt{34}}{5}\Big)^3-2\times\frac{5}{\sqrt{34}}}$ $=\frac{\frac{125\times15-2\times34\times34+10\times125}{125\sqrt{34}}}{\frac{125\times15+2\times34\times34-10\times125}{125\sqrt{34}}}$ $=\frac{1875-2312+1250}{1875+2312-1250}$ $=\frac{813}{2937} $ $=\frac{271}{979} $ Thus, $\frac{5\sin\theta-2\sec\theta+2\cos\theta}{5\sin\theta+2\sec\theta-2\cos\theta}=\frac{271}{979}$ Hence the correct option is $(a)$

M.C.Q (1 Marks)

GSEB - English Medium

If $3\cos\theta=5\sin\theta,$ then the value of $\frac{5\sin\theta-2\sec^3\theta+2\cos\theta}{5\sin\theta+2\sec^3\theta-2\cos\theta}$ is:

A$\frac{271}{979}$
B$\frac{316}{2937}$
C$\frac{542}{2937}$
D$\text{None of these}$

STD 10
Maths
Trigonometric Ratios
R.D. Sharma

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