Question

If α and β, are two angles in standard position in Quadrant II, find sin(α - β) for the given function values.

cos α = -3/5 and cot β = -12/5

Answer 1

Well... the root of everything here is the fact that\[\Large \sin(\alpha -\beta)= \sin(\alpha)\cos(\beta)-\cos(\alpha)\sin(\beta)\]

Answer 2

Now we just have to find the specific values for...\[\large \sin(\alpha) = \color{red}?\\\large \sin(\beta) = \color{red}?\\\large \cos(\alpha) = \color{red}?\\\large \cos(\beta) = \color{red}?\]

Answer 3

Good news is, we're told that the angles are in the second quadrant, so cosines there will always be... (positive/negative?) and sines will always be (positive/negative?)

^_^
I need you to work with me here :)

Answer 4

Sorry my internet is wack & was cutting on & off! Hold on, I'm reading what you wrote now:)

Answer 5

Take your time :)

Answer 6

Cosines will be negative & sines will be positive

Answer 7

cosa=-3/5 & cosb=-12/5 (I don't know how to make those signs?) right? because those values were given to us

Answer 8

Let's not get ahead of ourselves :)
cot is not the same as cos.

Answer 9

Oh wow.. I'm blind. So we only have cosa=-3/5

But can we do the sin = y/r, cos = x/r & cot = x/y rules? & substitute everything in?

Answer 10

Yup :)
Remember that \(\beta\) is in the second quadrant, so expect a negative value for x and a positive value for y.

Answer 11

Let me draw it out for you :)
Marked angle is \(\beta\).
can you mark out the values for x and y?

Answer 12

I honestly have no idea how to do that......

Answer 13

Well, they can be a lot of things, but let's stick to the simplest:
cot = x/y = -12/5
So what's a worthy candidate (one that's simple) for the value of x?

Answer 14

-12

Answer 15

Good :)
and for y?

Answer 16

5

Answer 17

See? Not that hard :P
I'd like to point out that cot would still be -12/5
even if x was -5 instead and y was 12.
However, that would put the angle in the fourth quadrant instead, and your questions specifies the second quadrant, so we need a negative value for x, and a positive one for y.

Answer 18

Now, use that brilliant thing called the pythagorean theorem to figure out this side...

Answer 19

13=c

Answer 20

That was quick :P
and 13 = r
let's be consistent, okay? :)
ET VOILA!
You now have a value for x, y, and r for your angle \(\beta\).
You can now find \(\sin(\beta)\) and \(\cos(\beta)\) with ease :)

Answer 21

sin=y/r = 5/13 cos=x/r = 12/13

Answer 22

I quote you:
"Cosines will be negative & sines will be positive"

Stick with the program.... tsk tsk XD

Answer 23

Or you could just notice that x here is not simply 12, but -12