Calculus help please?

Question

Just a product rule question

y= e^3x.sin(πx)

dy/dx = 3e^3xsin(πx) + e^3x.πcos(πx)

..........= 0

e^3x[3sin(πx) + πcos(πx)] = 0

e^3x ≠ 0 so 3sin(πx) + πcos(πx) = 0

or 3sin(πx) = -πcos(πx)

tan(πx) = -π/3

πx = 2.3314

x = 0.7427

Fred gave me this asnwer and i understood everything until tan(πx) = -π/3 this step. I get this step as well but using the calculator how do i proceed to the next two steps listed above.
Thanks
Arma

Answer 1

He was just shuffling things to the other side of the equation, to get all the "x" functions on one side and the constants on the other:

3sin(πx) = -πcos(πx)
sin(πx) = (-π/3)cos(πx)
sin(πx)/cos(πx) = -π/3
tan(πx) = -π/3

The next step is to take the inverse tangent:

tan^-1(tan(πx)) = tan^-1(-π/3)
πx = -0.8084488

That's the answer my calculator gives for the next step. However, tan(π + a) = tan(a), so we can add π to that answer and get another valid one:

πx + πn = -0.8084488, where n is an integer

If you set n=1, you get:

πx = 2.333144

Anyway, it all depends what your rules are for inverse tangents. You might be expected to produce all the possible answers, or you might be expected to use -π/2 to π/2, or you might be expected to use 0 to π.

Fred's answer was for the range 0 to π, which means he got a πx value that was π higher than the one my calculator gave.

Anyway, knowing that...

πx + πn = -0.8084488

... you can divide by pi:

x = -0.257337 - n (where n is any integer)

So your answer is probably -0.257337, or 0.742663 (the former plus one). Or if you need all the values, it's...

... -2.257337, -1.257337, -0.257337, 0.742663, 1.742663, 2.742663, ...

Answer 2

since : tan(πx) = -π/3
you use: tan^-1(-π/3) = πx
and get : x = [tan^-1(-π/3)]/π

where tan^-1 is tangent inverse on the calc

Answer 3

you see the value of -pi/3 from the caculator. itis
pi/3=1.047
now tan(pi x)=-1.047as it is negative, so pi x lies in the second quadrant..
pi x =180-46.31=133.69
x=133.69/pi=42.55 degrees. answer