SOLUTION: Given that sin(a) = 4 5 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) = 4 5 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 1204615: Given that
sin(a) =
4
5
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 math_tutor2020(3817) About Me  (Show Source):
You can put this solution on YOUR website!

Question: Given that sin(a) = 4/5 and cos(b) = -1/3, with a and b both in the interval [pi/2, pi], find the exact values of
sin(a+b)
and
cos(a-b)


The pythagorean trig identity is useful here.
sin%5E2%28x%29%2Bcos%5E2%28x%29+=+1
Use that identity to go from sin%28a%29+=+4%2F5 to cos%28a%29+=+-3%2F5
Angle 'a' is in quadrant 2 where cosine is negative.
I'll leave the steps and scratch work for the student to do.

If cos%28b%29+=+-1%2F3, then sin%28b%29+=+%282%2Asqrt%282%29%29%2F3 due to the pythagorean trig identity.
Sine is positive in quadrant Q2.

We'll then turn to this identity
sin%28a%2Bb%29+=+sin%28a%29cos%28b%29%2Bcos%28a%29sin%28b%29
which leads to
sin%28a%2Bb%29+=+%284%2F5%29%28-1%2F3%29%2B%28-3%2F5%29%2A%28%282%2Asqrt%282%29%29%2F3%29

sin%28a%2Bb%29+=+%28-4-6%2Asqrt%282%29%29%2F15

And,
cos%28a-b%29+=+cos%28a%29cos%28b%29%2Bsin%28a%29sin%28b%29

cos%28a-b%29+=+%28-3%2F5%29%2A%28-1%2F3%29%2B%284%2F5%29%2A%28%282%2Asqrt%282%29%29%2F3%29

cos%28a-b%29+=+%283%2B8%2Asqrt%282%29%29%2F15