Why do we need the Fourier transforms of sin and cos?
Yours is a far better question than it may seem on first reading.

The Fourier transform of a sine/cosine is a pair of Dirac's delta \$\delta\$ functions  at \$\pm f_c\$. Dirac's delta function can be thought as the continuous frequency equivalent of the discrete frequency coefficients of the Fourier series expansion.

The reason we mix them up in EE is because of analytical convenience. Look at the following example.

Suppose that, as part of a transmitting system, we have an modulator that takes a baseband signal \$x(t)\$ and modulates a carrier \$\cos(\omega_c t)\$ with it, so that the get a modulated signal \$s(t)=x(t)\cos(\omega_c t)\$.

If we then want to calculate the spectrum \$S(f)\$ of the modulated signal \$s(t)\$ we do need the Fourier transform of the carrier \$\cos(\omega_c t)\$. In this case:

$$
S(f) = \frac{1}{2} [X(f-f_c)+X(f+f_c)]
$$

where \$X(f)\$ is the Fourier transform (a.k.a. spectrum) of \$x(t)\$.

Thus, the baseband signal spectral components of \$x(t)\$ have been "moved up the frequency spectrum" around the carrier \$f_c\$. This result is the keystone of radio communications.

But how did we obtain the previous result, which looks deceivingly simple? It is obtained from the convolution of \$X(f)\$ with the Fourier transform of the carrier (the Dirac's delta pair we mentioned at the beginning). This straightforward calculation couldn't have been done without using the Fourier transform of the sine/cosine function. That's why we define it: because we need it.

In the end, the point is that in EE we usually use sine/cosine periodic functions along with aperiodic information-bearing signal, and we need to analyse them together within the same mathematical framework: the Fourier transform.