Identify the exact value of 157.5°

exact value of sin 157.5 without using a calculator

sin (157.5) = sin (315/2)

Identity: sin (x/2) = ±√[ (1-cosx) / 2]

select positive identity since 175 is in the 2nd quadrant where sin>0

sin (315/2) = √[ (1-cos315) / 2]

cos 315=cos45 in quadrant IV=√2/2

sin (315/2) = √[ (1-√2/2) / 2]=√[ (2-√2) / 4]=√ (2-√2) / 2

sin (157.5) = √ (2-√2) / 2

check using calculator:

sin157.5º≈0.382 ...

√ (2-√2) / 2≈0.382 ...