Finding the values of tan from a pair of given equations

**jashansinghal** · July 24th 2009, 02:06 AM

If

$\frac{\sin A }{\sin B}=p$

$\frac{\cos A}{\cos B}=q$

then find the values of $\tan A$ and $\tan B$

**SENTINEL4** · July 24th 2009, 02:46 AM

$tanA=\frac{sinA}{cosA}$ (1)

$\frac{sinA}{sinB}=p \Rightarrow sinA=psinB$ and
$\frac{cosA}{cosB}=p \Rightarrow cosA=pcosB$

Replace these two equations at equation (1) and you have

$tanA=\frac{sinA}{cosA}=\frac{psinB}{qcosB}$ , but it is $\frac{sinB}{cosB}=tanB$ so...
$tanA=\frac{p}{q}tanB$

**Soroban** · July 24th 2009, 05:57 AM

Hello, jashansinghal!

SENTINEL4 is correct.
And there is not enough information for a unique answer.

$\text{If }\:\begin{Bmatrix}{\color{blue}(1)} & \dfrac{\sin A }{\sin B}\:=\:p \\ \\[-3mm] {\color{blue}(2)} & \dfrac{\cos A}{\cos B}\:=\:q \end{Bmatrix}\quad \text{then find the values of }\tan A\text{ and }\tan B$

Divide ${\color{blue}(1)}$ by ${\color{blue}[2]}$ : . $\frac{\dfrac{\sin A}{\sin B}}{\dfrac{\cos A}{\cos B}} \;=\;\frac{p}{q} \quad\Rightarrow\quad \frac{\dfrac{\sin A}{\cos A}}{\dfrac{\sin B}{\cos B}} \;=\;\frac{p}{q}$ . $\Rightarrow\quad \frac{\tan A}{\tan B} \;=\;\frac{p}{q}$

We have the ratio of $\tan A$ and $\tan B$ . . . and that's the best we can do.

We can write: . $\begin{Bmatrix}\tan A\:=\:kp \\ \tan B \:=\:kq \end{Bmatrix}\;\;\text{for some real number }k \neq 0$

**Opalg** · July 24th 2009, 12:41 PM

Originally Posted by jashansinghal

If

$\frac{\sin A }{\sin B}=p$

$\frac{\cos A}{\cos B}=q$

then find the values of $\tan A$ and $\tan B$

Notice that $\sin A = p\sin B$ and $\cos A = q\cos B$ . But $\sin^2A+\cos^2A=1$ . Therefore $p^2\sin^2B + q^2\cos^2B = 1$ . Divide by $\cos^2B$ to get $p^2\tan^2B + q^2 = \sec^2B = 1 + \tan^2B$ , so that $(p^2-1)\tan^2B = 1-q^2$ . Hence $\boxed{\tan B = \pm\sqrt{\frac{1-q^2}{p^2-1}}}$ . (And as has already been noted, $\tan A = \tfrac pq\tan B$ ).

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