Precal trig help (Simplify)

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Trig formula is:
sin(A-B) = sinA * cosB-sinB * cosA


Basically my professor broke down like this
A B A B
cos45*sin30-cos30*sin45
Sin(45-30) = sin(15)

should it be this according to the trig formula?
B A A B
cos45*sin30-cos30*sin45
sin ( 30-45) = sin(-35)?

Your professor was referring to the A B A B you get when you write the formula with the Sines on the exterior. This way, you always use the A first.

You have: sin(A - B) = sinAcosB - sinBcosA
your professor has: sin (A - B) = sinAcosB - cosAsinB
These are the same since the last term can have the multiplication in any order.
sinBcosA = cosAsinB

So, when your professor was stating A B B A, the instructor wanted to help you remember.
The order of your trig functions in NOT sin cos sin cos in that form, but rather sin cos cos sin.
It appears your professor inadvertantly swithed the sin and cosines as it is written.
In solving sin(45 - 30), you would have A = 45, B = 30
sin45cos30 - cos45sin30
=(.71)(.866) - (.71)(.5)
=.259 = sin (15)
all of these decimals are rounded

When solving sin(30 - 45), the A and B are now reversed, so you would have
sin (30 - 45) A = 30, B = 45
sin30cos45 - cos30sin45
=(.5)(.71) - (.866)(.71)
= -.259 = sin(-15)


The numbers inside the parentheses at the end are coming from the subtraction in the problem's parentheses.

sin(45 - 30) = sin(15) since 45 - 30 = 15

The reason we expand the 15 out as a subtraction expression is to be able to use sin and cos of angles we know.

B has to be minus. B = 30o

The way you wrote it in the third line of your question is the way it should be written and substituted.

sin(A-B) = sinA * cosB-sinB * cosA

A = 45
B = 30

Sin(45 - 30) = Sin(15) = Sin45 * cos30 - sin30 * cos 45
Sin(15) = sqrt(2)/2 * sqrt(3)/2 - 1/2*sqrt(2)/2

The thing that you are overlooking (perhaps) is that the cos of a minus angle is usually considered plus, especially if it is less than 90o whereas the sine of a minus angle is minus. Check this with your calculator.

How did you get sin(−35) from sin(30−45) ? 30−45 = −15

Formula is sin(A−B) = sin(A)cos(B) − cos(A)sin(B)

What you have is:
cos(45) sin(30) − cos(30) sin(45)
= sin(30) cos(45) − cos(30) sin(45)
= sin(30−45)
= sin(−15)

Looks like your teacher incorrectly applied the formula. Your application is correct. However,

sin(30-45) = sin(-15)

AND, that can be simplified to -sin(15)

(sin(x) is an odd fcn)

sin(45 - 30)
= sin45 cos30 - sin30 cos45
= sin15