# How do you find the equation of the tangent line y=sinx at (pi/6, 1/2)?

Aug 31, 2017

$y - \frac{1}{2} = \frac{\sqrt{3}}{2} \left(x - \frac{\pi}{6}\right)$

#### Explanation:

The slope of the tangent line to a function $y$ at $x = a$ is found by calculating the value of $\frac{\mathrm{dy}}{\mathrm{dx}}$, the derivative of $y$, at $x = a$.

Where

$y = \sin x$

the derivative is given by

$\frac{\mathrm{dy}}{\mathrm{dx}} = \cos x$

The slope of the tangent line to $y$ at $x = \frac{\pi}{6}$ is found by evaluating the derivative of $y$ at $x = \frac{\pi}{6}$:

$m = \frac{\mathrm{dy}}{\mathrm{dx}} {|}_{x = \frac{\pi}{6}} = \cos \left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2}$

The slope of the tangent line is $\frac{\sqrt{3}}{2}$. Writing the equation of the line that passes through $\left(\frac{\pi}{6} , \frac{1}{2}\right)$ with slope $\frac{\sqrt{3}}{2}$ in point-slope form, we get:

$y - {y}_{1} = m \left(x - {x}_{1}\right)$

$y - \frac{1}{2} = \frac{\sqrt{3}}{2} \left(x - \frac{\pi}{6}\right)$

Aug 31, 2017

Differentiate y and evaluate $\frac{\mathrm{dy}}{\mathrm{dx}}$ at $x = \frac{\pi}{6}$

The equation of the tangent line would then be $y = a x + b$, where $a$ is the derivative at $x = \frac{\pi}{6}$ and $b$ can be solved by setting $y = \frac{1}{2} , x = \frac{\pi}{6}$

The equation would be $y = \frac{\sqrt{3}}{2} x + \frac{1}{2} - \frac{\sqrt{3} \pi}{12}$

#### Explanation:

Let the equation of the tangent line be $y = a x + b$

$\frac{\mathrm{dy}}{\mathrm{dx}} = \cos x$

$a = \cos \frac{\pi}{6} = \frac{\sqrt{3}}{2}$

$\therefore y = \frac{\sqrt{3}}{2} x + b$

$\frac{1}{2} = \frac{\sqrt{3}}{2} \cdot \frac{\pi}{6} + b$

$b = \frac{1}{2} - \frac{\sqrt{3} \pi}{12}$

Hence the equation of the tangent line is $y = \frac{\sqrt{3}}{2} x + \frac{1}{2} - \frac{\sqrt{3} \pi}{12}$

You can verify this answer visually too

graph{(y-sqrt(3)/2x-1/2+(sqrt(3)pi)/12)(y-sin(x))=0 [-1.259, 1.781, -0.477, 1.04]}

The reason the equation of a tangent line is as shown above is because in a linear function, $y = a x + b$, a represents the gradient / slope of the line and b represents the y-intercept.

By definition, the gradient of a tangent line is equal to the slope of a curve at the point where the tangent line meets the curve.

Hence, $a = \frac{\mathrm{dy}}{\mathrm{dx}}$ when $x = \frac{\pi}{6}$ and the rest of the equation can be derived through algebra