If 5tantheta-4=0, then the value of frac{5sintheta-4costheta}{5sintheta+4costheta} is:

Q: If $5\tan\theta-4=0,$ then the value of $\frac{5\sin\theta-4\cos\theta}{5\sin\theta+4\cos\theta}$ is:

Given that: $5\tan\theta-4=0.$We have to find the value of the following expression $\frac{5\sin\theta-4\cos\theta}{5\sin\theta+4\cos\theta}$ $$Since $5\tan\theta-4=0$ $\Rightarrow \tan\theta =\frac{4}{5}$ $\tan\theta=\frac{\text{Perpendicular}}{\text{Base}}$ We know that: $\Rightarrow \text{Base}=5$ $\Rightarrow{\text{perpendicular}}=4$ $\Rightarrow{\text{Hypotenuse=}\sqrt{\text{(Perpendicular)}^2+(\text{Base)}^2}}$ $\Rightarrow{\text{Hypotenuse=}}\sqrt{16+25}$ $\Rightarrow{\text{Hypotenuse=}}\sqrt{41}$ Since $\sin\theta =\frac{{\text{Perpendicular}}}{\text{Hypotenuse}}$ and $\cos\theta=\frac{\text{Base}}{{\text{Hypotenuse}}}$ Now we find $\frac{5\sin\theta-4\cos\theta}{5\sin\theta+4\cos\theta}$ $=\frac{5\times\frac{4}{\sqrt{41}}-4\times\frac{5}{\sqrt{41}}}{5\times\frac{4}{\sqrt{41}}+4\times\frac{5}{\sqrt{41}}}$ $=\frac{\frac{20}{\sqrt{41}}-\frac{20}{\sqrt{41}}}{\frac{20}{\sqrt{41}}+\frac{20}{\sqrt{41}}}$ $=0$ Hence the correct option is $(c)$

M.C.Q (1 Marks)

GSEB - English Medium

If $5\tan\theta-4=0,$ then the value of $\frac{5\sin\theta-4\cos\theta}{5\sin\theta+4\cos\theta}$ is:

A$\frac{5}{3}$
B$\frac{5}{6}$
C$\ 0$
D$\frac{1}{6}$

STD 10
Maths
Trigonometric Ratios
R.D. Sharma

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