1. finding trigonometric values

Can someone help me with this question?

Determine the exact value for each trigonometric expression, given sinA= 4/5, cos B= (-8/17),
[(pi)/2] < A < pi and (pi) < B < [3(pi)/2]

Find
1) sin (A-B)

2) Sin (A+B)

3) cos (A+B)

4) tan (A-B)

if you could just tell me one of the answers and how you got to it, i can probably figure out the rest. thank youuuuuu!!

2. Hello, clairez90!

Determine the exact value for each trigonometric expression,
. . given: . $\sin A = \frac{4}{5},\;\;\cos B = -\frac{8}{17},\qquad \frac{\pi}{2} \leq A \leq \pi\;\text{ and }\;\pi \leq B \leq \frac{3\pi}{2}$

We are given: . $\boxed{\sin A \:=\:\frac{4}{5}}\:=\:\frac{opp}{hyp}$
Using Pythagorus, we get: . $adj \:=\:\pm3 \quad\Rightarrow\quad \cos A \:=\:\pm\frac{3}{5}$

We are told that $A$ is in Quadrant 2, where cosine is negative.
. . Hence: . $\boxed{\cos A \:=\:-\frac{3}{5}}$
We have: . $\tan A \:=\:\frac{\sin }{\cos } \:=\:\frac{\frac{4}{5}}{\text{-}\frac{3}{5}} \quad\Rightarrow\quad \boxed{\tan A\:=\:-\frac{4}{3}}$

We are given: . $\boxed{\cos B \:=\:-\frac{8}{17}}\:=\:\frac{adj}{hyp}$
Using Pythagorus, we get: . $opp \:=\:\pm15\quad\Rightarrow\quad \sin B \:=\:\pm\frac{15}{17}$

We are told that $B$ is in Quadrant 3, where sine is negaitve.
. . Hence: . $\boxed{\sin B \:=\:-\frac{15}{17}}$
We have: . $\tan B \:=\:\frac{\sin B}{\cos B} \:=\:\frac{-\frac{15}{17}}{-\frac{8}{17}} \quad\Rightarrow\quad\boxed{ \tan B\:=\:\frac{15}{8}}$

Hence, we have: . $\boxed{\begin{array}{ccccccc}\sin A &=&\dfrac{4}{5} & & \sin B &=&\text{-}\dfrac{15}{17} \\ \\[-3mm]
\cos A &=&\text{-}\dfrac{3}{5} & & \cos B &=& \text{-}\dfrac{8}{17} \\ \\[-3mm]
\tan A &=&\text{-}\dfrac{4}{3} & & \tan B &=& \dfrac{15}{8} \end{array}}$

We will use these values in all of the following problems . . .

$1)\;\;\sin(A-B)$
We're expected to know: . $\sin(A \pm B) \:=\:\sin A\cos B \pm \sin B\cos A$

We have: . $\sin(A-B) \;=\;\left(\frac{4}{5}\right)\left(\text{-}\frac{8}{17}\right) - \left(\text{-}\frac{15}{17}\right)\left(\text{-}\frac{3}{5}\right) \;=\; \text{-}\frac{32}{85} - \frac{45}{85} \;=\;\boxed{-\frac{77}{85}}$

$2)\;\;\sin(A+B)$

$3)\;\;\cos(A+B)$
We should know: . $\cos(A \pm B) \;=\;\cos A\cos B \mp \sin A\sin B$

$4)\;\;\tan(A-B)$
We should know: . $\tan(A \pm B) \;=\;\frac{\tan A \pm \tan B}{1 \mp \tan A\tan B}$

3. thank you so much, Soroban!! that was extremely helpful!!!