SOLUTION: Given that sin(a) = 3 4 and cos(b) = − 1 3 , with a and b both in the interval 𝜋 2 , 𝜋 , find the exact values of sin(a + b) and cos(a − b)

Algebra ->  Trigonometry-basics -> SOLUTION: Given that sin(a) = 3 4 and cos(b) = − 1 3 , with a and b both in the interval 𝜋 2 , 𝜋 , find the exact values of sin(a + b) and cos(a − b)      Log On


   

Question 1204731: Given that
sin(a) =
3
4
and
cos(b) = −
1
3
,
with a and b both in the interval
𝜋
2
, 𝜋
,
find the exact values of
sin(a + b)
and
cos(a − b).
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!
given:
sin%28a%29+=3%2F4 and cos%28b%29+=+-1%2F3, with a and b+both in the interval pi%2F2, pi+=> Q+II
ln Q+II sine is positive and cosine is negative

sin%28a%29+=3%2F4=a%2Fc => a=3+and c=4
cos%28a%29=b%2Fc=b%2F4
using the Pythagorean Theorem, the missing side b=sqrt%284%5E2-3%5E2%29=sqrt%2816-9%29=sqrt%287%29

cos%28a%29=b%2Fc=sqrt%287%29%2F4
cos%28a%29=sqrt%287%29%2F4+or cos%28a%29=-sqrt%287%29%2F4+
we need +cos%28a%29=-sqrt%287%29%2F4+

and
cos%28b%29+=+-1%2F3, => b=-1, c=3
sin%28b%29=a%2F3
a=sqrt%283%5E2-%28-1%29%5E2%29=sqrt%289-1%29=sqrt%288%29=sqrt%282%2A4%29=2sqrt%282%29
sin%28b%29=%282sqrt%282%29%29%2F3+or sin%28b%29=-%282sqrt%282%29%29%2F3

we need sin%28b%29=%282sqrt%282%29%29%2F3


find the exact values of

sin%28a+%2B+b%29=cos%28b%29%2Asin%28a%29+%2B+cos%28a%29%2Asin%28b%29

sin%28a+%2B+b%29=-1%2F4-sqrt%2814%29%2F6


and

cos%28a+-b%29=cos%28a%29%2Acos%28b%29+%2B+sin%28a%29%2Asin%28b%29
cos%28a+-b%29=%28-sqrt%287%29%2F4%29%28-1%2F3%29%2B%283%2F4%29%28%282sqrt%282%29%29%2F3%29
cos%28a+-b%29=sqrt%287%29%2F12%2B1%2Fsqrt%282%29+
cos%28a+-b%29=sqrt%287%29%2F12%2Bsqrt%282%29%2F2