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

Oct 2014
133
2
Florida
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

Sin a-b.png

I just do not know how to do the sum formula for sine?

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

Please explain. Thank you!
 

skeeter

MHF Helper
Jun 2008
16,217
6,765
North Texas
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
 
Oct 2014
133
2
Florida
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?
 
Last edited:

skeeter

MHF Helper
Jun 2008
16,217
6,765
North Texas
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?
 
Oct 2014
133
2
Florida
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?
 
Last edited:

skeeter

MHF Helper
Jun 2008
16,217
6,765
North Texas
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.
 
Oct 2014
133
2
Florida
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.
 
Last edited:
Oct 2014
133
2
Florida
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
Jun 2008
16,217
6,765
North Texas
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
 
Mar 2016
1
0
Hoover
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...