This problem requires us to find out the equation of the tangent line, which has form y=mx+q.M is the angular coefficient or slope of the line and can be evaluated through the derivative of the function at the given point. Since g'(x)=d/dx(integral of cos(x*pi/3)/t) and the derivative is the inverse function of the integral , g'(x)=cos(x*pi/3)/t. Therefore g'(1)=cos(pi/3)=1/2. g'(1)=1/2=m.

Now we have to find q, which is the intersection point of the line with the y-axis. In order to do so we have to find a point through which the tangent passes: this point is the tangent point. To evaluate it we have to find the y-coordinate of g(1). g(1)=integral from 1 to 1 of cos(x*pi/3)/t , but indipendently of the integrand function, any inegral from 1 to 1 (or generally a to a) =0 (think of the area-definition of the integral). Therefore the tangent point is (1,0). The tangent has y=(1/2)x+q, substituting x=1 and y=0 we get that q=-1/2.