"How do you find the equation of the..." was answered by our community

Terrence Riddle

Answered question

2022-04-16

How do you find the equation of the line tangent to

y = \sin x

(\frac{π}{4}, \frac{\sqrt{2}}{2})

Mey9ci0

Beginner2022-04-17Added 14 answers

The slope of the tangent line can be found using the derivative of the function:

\frac{d y}{d x} = \cos x

The slope of the tangent line at

x = \frac{π}{4}

can then be found through plugging

\frac{π}{4}

into the derivative.

\cos (\frac{π}{4}) = \frac{\sqrt{2}}{2}

So, we have a line that passes through the point

(\frac{π}{4}, \frac{\sqrt{2}}{2})

and a slope of

\frac{\sqrt{2}}{2}

y - y_{0} = m (x - x_{0})

y - \frac{\sqrt{2}}{2} = \frac{\sqrt{2}}{2} (x - \frac{π}{4})

y = \frac{\sqrt{2}}{2} x - \frac{π \sqrt{2}}{8} + \frac{\sqrt{2}}{2}

y = \frac{4 \sqrt{2} x - π \sqrt{2} + 4 \sqrt{2}}{8}

Note that

\frac{\sqrt{2}}{8} = \frac{\sqrt{2}}{4 \sqrt{2} \sqrt{2}} = \frac{1}{4 \sqrt{2}} :

y = \frac{4 x - π + 4}{4 \sqrt{2}}

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