How do you find the equation of the tangent line y=sinx at (pi/6, 1/2)?
2 Answers
Explanation:
The slope of the tangent line to a function
Where
y=sinx
the derivative is given by
dy/dx=cosx
The slope of the tangent line to
m=dy/dx|_(x=pi/6)=cos(pi/6)=sqrt3/2
The slope of the tangent line is
y-y_1=m(x-x_1)
y-1/2=sqrt3/2(x-pi/6)
Differentiate y and evaluate
The equation of the tangent line would then be
The equation would be
Explanation:
Let the equation of the tangent line be
Hence the equation of the tangent line is
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,
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,