How many tangents can it adjust ?

Calculus Level 4

Find the number of tangents that are possible to the curve y = c o s ( x + y ) ; − 2 π ≤ x ≤ 2 π y=cos\left( x+y \right) \ ; \ -2\pi \leq x \leq 2\pi , which are parallel to the line x + 2 y = 0 x+2y=0 .

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The answer is 2.

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Sandeep Bhardwaj by Sandeep Bhardwaj , 28, India

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1 solution

Prakhar Gupta
Jan 12, 2015

We see that slope of the line must be − 1 2 \frac{-1}{2} . Diffrentiating the given equation we get. d y d x = − sin ⁡ ( x + y ) ( 1 + d y d x ) \dfrac{dy}{dx} = -\sin (x+y)(1+\dfrac{dy}{dx}) d y d x = − sin ⁡ ( x + y ) 1 + sin ⁡ ( x + y ) \dfrac{dy}{dx}=\dfrac{-\sin (x+y)}{1+\sin (x+y)} 1 2 = − sin ⁡ ( x + y ) 1 + sin ⁡ ( x + y ) \dfrac{1}{2}=\dfrac{-\sin (x+y)}{1+\sin (x+y)} 1 + sin ⁡ ( x + y ) = 2 sin ⁡ ( x + y ) 1+\sin(x+y)=2\sin(x+y) 1 = sin ⁡ ( x + y ) 1=\sin(x+y) cos ⁡ ( x + y ) = 0 \cos(x+y)=0 y = 0 y=0 sin ⁡ ( x ) = 1 \sin(x)=1 In the range − 2 π ≤ x ≤ 2 π -2\pi\leq x\leq 2\pi we have 2 values of x where sin ⁡ ( x ) = 1 \sin(x) = 1 .

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