91Ó°ÊÓ

Consider the following conditions of integral to find the angle between f and g.

It will check three integrals.

1. 
   $\begin{array}{rcl}\underset{-\mathrm{π}}{\overset{\mathrm{π}}{∫}}{\cos }^{2}tdt& =& \underset{-\mathrm{π}}{\overset{\mathrm{π}}{∫}}\frac{\cos \left(2t\right)+1}{2}dt\\ & =& \frac{1}{2}×2×\underset{0}{\overset{\mathrm{π}}{∫}}\left(\cos \left(2t\right)+1\right)dt\\ & =& \mathrm{π}\end{array}$
2. 
   $\begin{array}{rcl}\underset{-\mathrm{π}}{\overset{\mathrm{π}}{∫}}{\cos }^{2}tdt& =& \underset{-\mathrm{π}}{\overset{\mathrm{π}}{∫}}\frac{1-\cos \left(2t\right)}{2}dt\\ & =& \frac{1}{2}×2×\underset{0}{\overset{\mathrm{π}}{∫}}\left(1-\cos \left(2t\right)+1\right)dt\\ & =& \mathrm{π}\end{array}$
3. 
   $\begin{array}{rcl}\underset{-\mathrm{π}}{\overset{\mathrm{π}}{∫}}\sin t\cos tdt& =& \underset{-\mathrm{π}}{\overset{\mathrm{π}}{∫}}\frac{\sin 2t}{2}dt\\ & =& 0\end{array}$
   

Now, observe the following

Also, the length of the functions f and g can be found as,

Therefore, the angle between f and g

Hence, the value of the angle between f and g will be

$∠\left(f,g\right)=\mathrm{δ}$