We will need the values of cos(s) and cos(t)cos(s)andcos(t), therefore, we shall use the identity:
cos(x) = +-sqrt(1-sin^2(x))" [1]"cos(x)=±√1−sin2(x) [1]
Substitute sin(s)= -12/13sin(s)=−1213 into equation [1]:
cos(s) = +-sqrt(1-(-12/13)^2)cos(s)=±√1−(−1213)2
cos(s) = +-sqrt(169/169-144/169)cos(s)=±√169169−144169
cos(s) = +-sqrt(25/169)cos(s)=±√25169
cos(s) = +-5/13" [2]"cos(s)=±513 [2]
We are told that ss is in the fourth quadrant, therefore, we shall choose the positive value:
cos(s) = 5/13cos(s)=513
Substitute sin(t)= 4/5sin(t)=45 into equation [1]:
cos(t) = +-sqrt(1-(4/5)^2)cos(t)=±√1−(45)2
cos(t) = +-sqrt(25/25-16/25)cos(t)=±√2525−1625
cos(t) = +-sqrt(9/25)cos(t)=±√925
cos(t) = +-3/5cos(t)=±35
We are told that tt is in the second quadrant, therefore, we shall choose the negative value:
cos(t) = -3/5" [3]"cos(t)=−35 [3]
Using the identity,
cos(s+t) = cos(s)cos(t) - sin(s)sin(t)cos(s+t)=cos(s)cos(t)−sin(s)sin(t)
, we substitute cos(s) = 5/13, cos(t) = -3/5, sin(s) = -12/13, and sin(t) = 4/5cos(s)=513,cos(t)=−35,sin(s)=−1213,andsin(t)=45:
cos(s+t) = (5/13)(-3/5) - (-12/13)(4/5)cos(s+t)=(513)(−35)−(−1213)(45)
cos(s+t) = 33/65cos(s+t)=3365
Using the identity,
cos(s-t) = cos(s)cos(t) + sin(s)sin(t)cos(s−t)=cos(s)cos(t)+sin(s)sin(t)
, we substitute cos(s) = 5/13, cos(t) = -3/5, sin(s) = -12/13, and sin(t) = 4/5cos(s)=513,cos(t)=−35,sin(s)=−1213,andsin(t)=45:
cos(s-t) = (5/13)(-3/5) + (-12/13)(4/5)cos(s−t)=(513)(−35)+(−1213)(45)
cos(s-t) = -63/65cos(s−t)=−6365
