Find the values of x on the curve below at which the tangent is horizontal. Use n as an arbitrary integer.

Question

Find the values of x on the curve below at which the tangent is horizontal. Use n as an arbitrary integer.

y = sin(x) / 2 + cos(x)

x=__________

Answer 1

y = sinx / 2 + cosx

The tangent line is horizontal when y' = 0

So, cosx / 2 - sinx = 0

cosx / 2 = sinx

tanx = 1/2

Tangent is positive in quadrants 1 and 3.

x = Tan^-1(1/2) + nπ = 0.4636 + nπ