Is the sin(3pi/4) equal to sin(pi/4)?

Question

Which is to say, is this a true statement

 

 sin(3pi/4)=sin(pi/4),

 

or do the individual trig functions merely have the same value?

 

 

I know it seems like a strange distinction, but it is the source of lively debate between me and a friend. 

 

 

 Similarly, is arcsin(sin(3pi/4))=arcsin(sin(pi/4)? 

Answer 1

Yes, because all of the possible positive sine values of the first quadrant are replicated in the second quadrant (but in reverse order). The understanding of this is enhanced by study of the UNIT CIRCLE, where the length of arc subtended by an angle is actually the measure of the arc, in radians.  The 'reference triangle' has vertical coordinate that is the sine, and horizontal coordinate that is the cosine, of that angle in standard position. For this reason, cosines are negative in quadrants II and II, and sines are negative in quadrants III & IV.