First, we'll convert the angle into degrees. We'll do so by replacing pi by 180, as following:
[tex]\frac{19\pi}{4}\rightarrow\frac{19\cdot180}{4}\rightarrow855[/tex]
This way, we'll get that the angle is 855°. To this angle, we'll substract the nearest factor of 360: 720
[tex]855-720=135[/tex]
This way, we get an angle we can work on (between 0° and 360°)
Now, notice that 135° is an angle that belongs to the second quadrant. Because of this, we'll habe to substract 90° to get the reference angle:
[tex]135-90=45[/tex]
We get that the reference angle is 45°. Now, let's switch this angle back into radians. To do so, we multiplty by pi and divide by 180, as following:
[tex]45\cdot\frac{\pi}{180}=\frac{\pi}{4}[/tex]
This way, we'll have that the reference angle for
[tex]\frac{19\pi}{4}[/tex]
is:
[tex]\frac{\pi}{4}[/tex]
And since cosine is negative in the second quadrant (where the original angle belongs), we can conclude that:
[tex]\cos (\frac{19\pi}{4})=-\cos (\frac{\pi}{4})[/tex]
And we'll have that:
[tex]-\cos (\frac{\pi}{4})=-\frac{\sqrt[]{2}}{2}[/tex]
