SOLUTION: This question is ridiculously confusing to me, any help is greatly appreciated! If it is known that sin a = 4/5, pi/2 < a < pi and that sin b = - (2 * sqrt(5) / 5), pi < b <

Algebra ->  Trigonometry-basics -> SOLUTION: This question is ridiculously confusing to me, any help is greatly appreciated! If it is known that sin a = 4/5, pi/2 < a < pi and that sin b = - (2 * sqrt(5) / 5), pi < b <       Log On


   

Question 868504: This question is ridiculously confusing to me, any help is greatly appreciated!
If it is known that sin a = 4/5, pi/2 < a < pi
and that sin b = - (2 * sqrt(5) / 5), pi < b < 3pi/2,
find the exact value of cos( a + b )
Pretty please help me out, i'm in a crunch right now :/ thanks though :)
Answer by Fombitz(32388) About Me  (Show Source):
You can put this solution on YOUR website!
From trig relation,
cos%28a%2Bb%29=cos%28a%29cos%28b%29-sin%28a%29sin%28b%29
So you need to find cos%28a%29 and cos%28b%29.
Since cos%5E2%28a%29%2Bsin%5E2%28a%29=1
then,
cos%5E2%28a%29%2B%284%2F5%29%5E2=1
cos%5E2%28a%29=25%2F25-16%2F25
cos%5E2%28a%29=9%2F25
cos%28a%29=0+%2B-+3%2F5
Since you're given information that pi%2F2%3Ca%3Cpi, then cos%28a%29 must be negative (2nd quadrant) so you choose,
cos%28a%29=-3%2F5
Now for b.
Same thing,
cos%5E2%28b%29%2Bsin%5E2%28b%29=1
cos%5E2%28b%29%2B%28-%282sqrt%285%29%29%2F5%29%5E2=1
cos%5E2%28b%29%2B%284%2F5%29=1
cos%5E2%28b%29=5%2F5-4%2F5
cos%5E2%28b%29=1%2F5
cos%28b%29=0+%2B-+sqrt%285%29%2F5
From the information, b lies in the 3rd quadrant, where the cosine is also negative
cos%28b%29=-sqrt%285%29%2F5
Now go back to the original equation and plug in the values.
cos%28a%2Bb%29=cos%28a%29cos%28b%29-sin%28a%29sin%28b%29

cos%28a%2Bb%29=%283sqrt%285%29%29%2F25%2B%288sqrt%285%29%29%2F25%29
cos%28a%2Bb%29=%2811sqrt%285%29%29%2F25
.
.
.
As a check using inverse trig functions,
a=126.9 degrees
b=243.4 degrees
a%2Bb=370.3 degrees
cos%28370.3%29=0.9839
11%2Asqrt%285%29%2F25=0.9839