find cos(s+t) and cos(s-t) using cosine of a sum and cosine of a difference identities, if sin(s)=2/3 in quadrant 2 and sin (t)=-1/3 in quadrant 4

Question

use sum and difference identities to find cos(s+t) and cos(s-t)

Answer 1

cos(s+t)=cos(t)cos(s)-sin(t)sin(s)\

=√(1-sin²t)*-√(1-sin²s)-sin(t)sin(s) ,,,, choose + for first square root since t is in quadrant 4 and choose minus sign for second square root since s is in quadrant 2

=

-√(40/81)+2/9,

cos(s-t)=cos(t)cos(s)+sin(t)sin(s)

=

(1-sin²t)*-√(1-sin²s)+sin(t)sin(s)

=-√(40/81)-2/9,

Answer 2

sins = 2/3 in quadrant 2

s = 138.2 degrees

sint = -1/3 in quadrant 4

t = 340.53 degrees

s+t = 138.2+ 340.53 = 478.73

cos478.73 =cos118.73 =- .481

s-t = 138.2 - 340.53 = -202.33

cos-202.33 =cos157.67 =-.925

if you use the sum and difference formulas you should get close to the above answers

the above calculations, using inverse trig functions on a calculator are a check on the answers below

if sins = 2/3 in quadrant 2, then coss = -(sqr9-4)/3 = -sqr5/3

if sint = -1/3 in quadrant 4, then cost = sqr8/3

plug those values into the following formulaa

cos(s+t) = cosscost - sinssint = -sqr5/3)(sqr8/3) - 2/3)(-1/3) = -sqr40/9 +2/9 =about -.481

cos(s-t) = cosscost + sinssint = -sqr5/3(sqr8/3) + (2/3)(-1/3) =-sqr40/9 - 2/9 = about -.925