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!
