Math help... desperately needed...?

Question

Find the exact value of

1 - sin² 165°

show all working out..

thanks you..

much appreciated..

Answer 1

1 - sin² 165° = cos² 165° (BY sin² A+cos² A = 1 )

to find exact, we must use sum formula
cos(A+B) = cosAcosB - sinAsinB
cos(165°) = cos(120°+45°)
=cos120°cos45° - sin120°sin45°
=(-1 / 2)(sqrt2/2) - (sqrt3/2)(sqrt2 / 2)
= - (sqrt2 + sqrt6) / 4
now square this cos(165°)
= [- (sqrt2 + sqrt6) / 4]²
= (2 + 2sqrt12 + 6) / 16
= (8 + 2sqrt12) / 16 ===> since sqrt12=2sqrt3
=(8 + 4sqrt3) / 16




Double check this answer by working back.
You should get the correct answer.

Answer 2

1 - sin² 165° = cos² 165° (BY sin² A+cos² A = 1 )

to find exact, we must use sum formula
cos(A+B) = cosAcosB - sinAsinB
cos(165°) = cos(120°+45°)
=cos120°cos45° - sin120°sin45°
=(-1 / 2)(sqrt2/2) - (sqrt3/2)(sqrt2 / 2)
= - (sqrt2 + sqrt6) / 4
now square this cos(165°)
= [- (sqrt2 + sqrt6) / 4]²
= (2 + 2sqrt12 + 6) / 16
= (8 + 2sqrt12) / 16 ===> since sqrt12=2sqrt3
=(8 + 4sqrt3) / 16

______________________________
check answer:
DEGREE MODE on alculator
type in my answer =(8 + 4sqrt3) / 16 = .9330127019..
type in your original 1 - sin² 165° = .9330127019....
YEP! exactly the same!
=]
bravo!


...

Answer 3

sin²Θ + cos²Θ = 1 so
1-sin²Θ = cos²Θ or
1-sin²165 = cos² 165 = .933

HTH

Doug