SOLUTION: if sin s = -1/4 and s is in quadrant 4 and if cos t = -2/5 and t is in quadrant 2, find the following: cos(s + t) sin(s - t)

Algebra ->  Trigonometry-basics -> SOLUTION: if sin s = -1/4 and s is in quadrant 4 and if cos t = -2/5 and t is in quadrant 2, find the following: cos(s + t) sin(s - t)      Log On


   

Question 735057: if sin s = -1/4 and s is in quadrant 4 and if cos t = -2/5 and t is in quadrant 2, find the following:
cos(s + t)
sin(s - t)
Answer by lwsshak3(11628) About Me  (Show Source):
You can put this solution on YOUR website!
if sin s = -1/4 and s is in quadrant 4 and if cos t = -2/5 and t is in quadrant 2, find the following:
cos(s + t)
sin(s - t)
***
let O=opposite side of reference right triangle
let A=adjacent side of reference right triangle
let H=hypotenuse of reference right triangle
sin s=-1/4 (in Q4)=O/H
O=-1, H=4
A=sqrt%28H%5E2-O%5E2%29=sqrt%284%5E2-1%5E2%29=sqrt%2816-1%29=sqrt%2815%29
cos%28s%29=A%2FH=sqrt%2815%29%2F4
..
cos t = -2/5(in Q2)=A/H
A=-2, H=5
O=sqrt%28H%5E2-A%5E2%29=sqrt%285%5E2-2%5E2%29=sqrt%2825-4%29=sqrt%2821%29
sin%28t%29=O%2FH=sqrt%2821%29%2F5
..
cos(s + t)=cos s cos t-sin s sin t
=√15/4*-2/5-(-1/4)*√21)/5
=-2√15/20+√21/20
=(√21-2√15)/(20)
..
sin(s-t)=sin s cos t-cos s sin t
=-1/4*-2/5-√15/4*√21/5
=2/20-√315/20
=(2-√315)/20
..
Check with calculator:
sin(s)=-1/4(in Q4)
s≈345.52º
cos(t)=-2/5 (in Q2)
t≈113.58
..
s+t≈459.1
reference angle=80.9º(in Q2 where cos<0)
Cos(s+t)=cos(99.1)≈-0.1581..
(√21-2√15)/20≈-0.1581..
..
(s-t)≈231.94
reference angle≈51.94 (in Q3 where sin<0)
sin(s-t)=sin(51.94)≈-0.7874...
(2-√315)/20≈-0.7874...