Question

What angle does the tangent to the curve y(x)=sin^2 (x/4)+12sin(x/4) at x=4_ make with the x-axis?

Express your answer to 3 significant figures. Choose the smaller angle, i.e., your answer should be positive and less than 90 degrees.

Answer 1

y(x)=sin^2 (x/4)+12sin(x/4)

y(x) = sin^2(x/7) +16sin(x/7)

From the Chain Rule,

y'(x) = 2sin(x/4)*(1/4)cos(x/4) + 12(1/4)cos(x/4).

At x=4pi, the slope of the tangent line is

y'(4pi) = 2sin(4pi/4)*(1/4)cos(4pi/4) + 12(1/4)cos(4pi/4).

= 2sin(pi)*(1/4)cos(pi) + 12(1/4)cos(pi)

= 2(0)(1/4)(-1) + 12(1/4)(-1)

= -3

The angle (in radians) that the tangent line makes to the positive x-axis is

pi - arctan(3)= 1.249 rad =71.565 degree