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find the equation of the line tangent to the graph of sin(x) at the following value of...

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user7230927 | (Level 1) Salutatorian

Posted June 26, 2013 at 4:56 PM via web

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find the equation of the line tangent to the graph of sin(x) at the following value of x:

`x=4pi`

 

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sciencesolve | Teacher | (Level 3) Educator Emeritus

Posted June 26, 2013 at 5:20 PM (Answer #1)

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You should find the equation of the tangent line, to the graph of function `y = sin x` , at the point `x = 4pi` , such that:

`y - sin(4pi) = (dy)/(dx)|_(x = 4pi)(x - 4pi)`

You need to evaluate ` sin 4pi` using the following trigonometric identity, such that:

`sin 4pi = sin 2*(2pi) = 2 sin 2pi*cos 2pi`

Since `sin 2pi = 0` yields:

`sin 4pi = 0`

You need to evaluate the derivative of the function `y = sin x` such that:

`(dy)/(dx) = cos x => (dy)/(dx)|_(x = 4pi) = cos 4pi = cos^2 2p` i

Since `cos 2pi = 1` yields:

`cos 4pi= 1`

Replacing 0 for `sin 4pi` and 1 for `cos 4pi` yields:

`y - 0 = 1*(x - 4pi) => y = x - 4pi`

Hence, evaluating the equation of the tangent line to the graph of function `y = sin x` , at the point `x = 4pi` , yields `y = x - 4pi.`

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