# finding trigonometric values

• May 12th 2008, 01:30 PM
clairez90
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!! :)
• May 12th 2008, 02:44 PM
Soroban
Hello, clairez90!

Quote:

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

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

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

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

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

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

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

Quote:

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

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

Quote:

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

Quote:

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

Quote:

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

• May 12th 2008, 04:02 PM
clairez90
thank you so much, Soroban!! that was extremely helpful!!! :)