The equation of tangent line is "[tex]3x+y = \frac{7 \pi}{4}[/tex]".
Given equation is:
→ [tex]y \ sin 12x = x \ cos 2y[/tex]
The points,
→ [tex](\frac{\pi}{2} , \frac{\pi}{4} )[/tex]
By differentiating with respect to "x", we get
→ [tex]y(12 \ cos 12x) +sin 12x\frac{dy}{dx} = x(-2sin2y \fracdy}{dx} )+cos 2y[/tex]
At [tex](\frac{\pi}{2}, \frac{\pi}{4} )[/tex],
→ [tex]\frac{\pi}{4} (12 cos 12\frac{\pi}{2} )+sin 12 \frac{\pi}{2} \frac{dy}{dx}= \frac{\pi}{2} (12sin 2\frac{\pi}{2} \frac{dy}{dx} ) +cos 2\frac{\pi}{4}[/tex]
→ [tex]\frac{\pi}{4} (12 cos 6 \pi)+sin 6 \pi \frac{dy}{dx} = - \pi \frac{dy}{dx} +0[/tex]
→ [tex]3x+0= -\pi \frac{dy}{dx}[/tex]
→ [tex]\frac{dy}{dx} =-3[/tex]
hence,
The tangent line equation will be:
→ [tex](y-\frac{\pi}{4} ) = -3(x - \frac{\pi}{2} )[/tex]
[tex]3x+y = \frac{\pi}{4} +\frac{3 \pi}{2}[/tex]
[tex]3x+y = \frac{7 \pi}{4}[/tex]
Thus the above approach is the appropriate one.
Learn more about tangent here:
https://brainly.com/question/1600023
