The expression #sin(-(25pi)/4)# may look daunting, but consider that #sin(-(25pi)/4)=sin(-(((3times8)+1)pi)/4)#.
Since there are #2pi# radians in a circle this means we make three complete rotations about the unit circle in the clockwise direction, and another #pi/4# beyond that.
So, because of this we can say that #sin(-(25pi)/4)=sin(-pi/4)# . If you remember your unit circle, you will recall that #sin(pi/4)=1/sqrt(2)#. Since we are rotating in the clockwise direction, and the sine function corresponds to our y-values, we know that we end up below the x-axis when our rotating is done, so #sin(-pi/4)=-1/sqrt(2)#
The Pythagorean Theorem says
#sin^2(x)+cos^2(x)=1#
So,
#cos(-pi/4)=sqrt(1-(-1/sqrt(2))^2)=sqrt(1-1/2)=sqrt(1/2)=1/sqrt(2)#
Also, since this is in the 4th quadrant (at about 5 o'clock) cosine is positive.
Finally,
#tan(x)=sin(x)/cos(x)#
so,
#tan((-25pi)/4)=sin((-25pi)/4)/cos((-25pi)/4)=sin(-pi/4)/cos(-pi/4)#
#=(-1/sqrt(2))/(1/sqrt(2))=(-1/cancelsqrt(2))/(1/cancelsqrt(2))=-1#
No calculator needed!
