3 years ago
Use half-angle identities to find an exact value without a calculator.
cos 75°
1 Answers
Best Answer
Ellie90 Staff answered 3 years ago
Ellie90 Staff answered 3 years ago
Let 75°=(150°)/(2):
cos 75°=cos (150°)/(2)
Use the half-angle identity for sine sin (u)/(2)=pm sqrt((1+cos u)/(2)) where u = 150°. Since 75° is in Ql, then we take the positive identity:
=sqrt((1+cos 150°)/(2))
By special angles, cos 150°=-(sqrt(3))/(2)
=sqrt((1+(-(sqrt(3))/(2)))/(2))
=sqrt(((2)/(2)-(sqrt(3))/(2))/(2))
=sqrt(((2-sqrt(3))/(2))/(2))
=sqrt((2-sqrt(3))/(4))
=(sqrt(2-sqrt(3)))/(2)
Result:
(sqrt(2-sqrt(3)))/(2)