Understanding how to express basis with trig functions

Problem: create the basis (f1,f2) by rotating
The orthonormal basis vectors by the
Angle 3pi/4 in r^2. We have basis e aswell as
Standard,basis

Ok im on My iphone unable to draw any picture
But imagine urself with (e1,e2)
E1 is pointing in x direction and e2 pointing straight
Up on y

I Will get to My question soon.
Ok we create another basis f1,f2 and rotate
It 3pi/4

I guess now F1 is pointing 135 degrees and
F2 will get onto 225 degrees, correct me
If i think wrong

According to the answer

We get this:

F1= -e1*cos 3pi/4 + e2*sin 3pi/4
F2= -e1*sin 3pi/4 - e2*cos 3pi/4

I understand nothing of this
I would like To know how they picked
Cos,sine for f1 and how they chose
Sin,cos for f2, how should one think
When it comes to vectors? Is there any
General rule I should think of while selecting
The trig functions

Uhmm I completed calculus a while back, but
I have pretty bad skills in trig

Hope someone can figure it out

Yes, there's a formula (a 2x2 matrix with cosines on the diagonal, sine and -sine off the diagnoal), but you should know how to get that result. Memorizing this can be tricky, because it's easy to put the negative sine in the wrong matix, depending on the direction of the angle of rotation and which bases are the domain and which are the range of that matrix multiplication.

This is all about drawing a picture, and breaking down your new (just rotated) unit vectors into compinetns in terms of the original unit vectors. You'll have right triangle with hypotenuse of length 1, and an angle in the triangle. Then use sine = opposite/hypotenuse, cosine = adjacent/hypotenuse to get those coeficients. Then the thing to watch for is the +/- signs. This is the kinda thing that, after you do a few, seems pretty simple. Watching if the signs make sense is the only part that requires any great care.

For me, I check my work by comparing it to the standard example of rotating the standard basis vectors in the x and y directions counter clockwise by an angle less than 90 degrees. The new basis vectors should have all of their signs positive, except the x-coordinate of the rotated y-direction vector.

I am eager to know more in detail how it's done, could u or someone do it by the formula
Or draw a picture and point it out

Thanks

The formula for a rotation in the plane can be found everywhere, but to understand it this really requires a diagram. The thing to do is to direct you to a link that has it. I did some googling, but didn't find the nice diagram that shows why the formula is what it is in terms of triangles and trigonometry. That kinda shocked me - it must be out there, somewhere on the web.
Maybe you'll have better luck. I'd suggest search terms like: 2x2 rotation plane derivation.
Sorry I can't be of more help.

UPDATE:
I drew some pictures in Microsoft Paint. See if these make it any clearer (look at them in sequence):

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