(Centripetal Force) Why is this not correct?

[TheExibo]

There is a banked curve problem asking for the correct angle to keep the car on the road:

http://imgur.com/Ni3j6dB

My teacher said the following equation must be used to find normal force:

F(normal) = F(gravity)/cosθ

Why is it, that you cannot find the x-component of the normal force, by finding normal force with

F(gravity)cosθ = F(normal)

and then using sinθ to find the x-component?

 

[_N3WTON_]

There is a banked curve problem asking for the correct angle to keep the car on the road:

http://imgur.com/Ni3j6dB

My teacher said the following equation must be used to find normal force:

F(normal) = F(gravity)/cosθ

Why is it, that you cannot find the x-component of the normal force, by finding normal force with

F(gravity)cosθ = F(normal)

and then using sinθ to find the x-component?

The equation your teacher gave you is correct, in the situation you are describing (trying to find the "banking angle") the normal force will be greater (or perhaps equal to) the force due to gravity, so your equation is incorrect...

 

[TheExibo]

Now it makes sense, normal force is greater than the cos of geavity. Thanks!

 

[_N3WTON_]

Now it makes sense, normal force is greater than the cos of geavity. Thanks!

Note that they would be equal if:
\cos(\theta) = 1
ie, the surface was flat...

 

[haruspex]

Now it makes sense, normal force is greater than the cos of geavity. Thanks!

Yes, but that only demonstrates your equation is wrong, it doesn't explain why it is wrong.
Your equation is presumably obtained by resolving forces normal to the plane, but the resultant (the centripetal force) has a component in that direction, so the full equation would be $N+F_c \sin(\theta) = mg \cos(\theta)$.