# Find sin(a - b) if sin(a)= 5/13 where a is in the second quadrant and cos(b)= -7/25 w

#### realbrandon

1. Find sin(a - b) if sin(a)= 5/13 where a is in the second quadrant and cos(b)= -7/25 where b is in the third quadrant.

2. Find cos(a + b) if sin(a)= 8/17 where a is in the first quadrant and tan(b)= -7/24 where b is in the second quadrant.

This is what I got for number 1:

sinα = 5/13 cos²α = 1-sin²α = 144/169 Since α is in the second quadrant, cosα = -√(144/169) = -12/13.

cosβ = -7/25 sin²β = 1-cos²β = 576/625 Since β is in the third quadrant, sinβ = -√(576/625) = -24/25 I just do not know how to do the sum formula for sine?

...and I am not sure how to do number 2?

#### skeeter

MHF Helper
Re: Find sin(a - b) if sin(a)= 5/13 where a is in the second quadrant and cos(b)= -7/

I just do not know how to do the sum formula for sine?
it's actually the difference formula. you're on the right track for #1 ...

$\sin(a-b) = \sin{a}\cos{b} - \cos{a}\sin{b}$

plug in the values and do the arithmetic

#### realbrandon

Re: Find sin(a - b) if sin(a)= 5/13 where a is in the second quadrant and cos(b)= -7/

Like this sin(-12/13) * cos(-24/25) - cos(-12/13) * sin(-24/25)? because I get 0.0369 and I do not think that is a choice? Would the answer be the last one 315/325 since it is the only positive one and the second quadrant is positive?

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#### skeeter

MHF Helper
Re: Find sin(a - b) if sin(a)= 5/13 where a is in the second quadrant and cos(b)= -7/

Like this sin(-12/13) * cos(-24/25) - cos(-12/13) * sin(-24/25)? because I get 0.0369 and I do not think that is a choice?
no ... and put away the calculator ...

$\color{red}{\sin{a}}\color{blue}{\cos{b}} - \color{green}{\cos{a}} \color{orange}{\sin{b}}$

$\color{red}{\left(\dfrac{5}{13}\right)} \color{blue}{\left(-\dfrac{7}{25}\right)} - \color{green}{\left(-\dfrac{12}{13}\right)} \color{orange}{\left(-\dfrac{24}{25}\right)}$

get the idea?

#### realbrandon

Re: Find sin(a - b) if sin(a)= 5/13 where a is in the second quadrant and cos(b)= -7/

Ok I got it -323/325. Now do I do the same thing for number 2?

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#### skeeter

MHF Helper
Re: Find sin(a - b) if sin(a)= 5/13 where a is in the second quadrant and cos(b)= -7/

Ok I got it -323/325. Now do I do the same thing for number 2?
yes, use the sum formula for cosine.

#### realbrandon

Re: Find sin(a - b) if sin(a)= 5/13 where a is in the second quadrant and cos(b)= -7/

Which would be this cos(a + b) = cos A cos B - sin A sin B? How do I get cos B and why does the question have tan?

Find cos(a + b) if sin(a)= 8/17 where a is in the first quadrant and tan(b)= -7/24 where b is in the second quadrant.

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#### realbrandon

Re: Find sin(a - b) if sin(a)= 5/13 where a is in the second quadrant and cos(b)= -7/

I think I got it (15/17)(-24/25) - (8/17)(7/25) = -416/425?

#### skeeter

MHF Helper
Re: Find sin(a - b) if sin(a)= 5/13 where a is in the second quadrant and cos(b)= -7/

I think I got it (15/17)(-24/25) - (8/17)(7/25) = -416/425?
correct

#### prestwig1

Re: Find sin(a - b) if sin(a)= 5/13 where a is in the second quadrant and cos(b)= -7/

5/12/13 and 7/24/25 are both Pythagorean triples 8/15/17 is a Pythagorean triple as well. Sin(a-b)= sin(a)cos(b) - cos(a)sin(b) Since we are dealing with Pythagorean triples all the values are known. The rest is just arithmetic. In second quadrant cos is negative sign is positive. In third quadrant both are negative. In first quadrant both are positive...