# If sin B = -12/13 then what is cos 2B ?

Dec 2, 2016

$\sin \text{B} = \frac{12}{13}$
$\text{B} = {\sin}^{-} 1 \left(- \frac{12}{13}\right)$

#### Explanation:

Once you have worked out the value of $\text{B}$, you simply have to type $\cos \left(2 \text{B}\right)$ in your calculator to get the answer. However, this will give you the principle value of $2 \text{B}$. To find out the value of $2 \text{B}$ in the third quadrant, do $360 - \cos \left(2 \text{B}\right)$.

The number of significant figures/decimal places is up to you.
Another tip is that if you want a more accurate answer you can type $\cos \left(2 \times {\sin}^{-} 1 \left(- \frac{12}{13}\right)\right)$ into your calculator. This will help you to avoid rounding errors. Then, you can do $360 - \text{ANS}$ (instead of rounding).

Dec 3, 2016

$\cos 2 B = - \frac{119}{169}$

#### Explanation:

Given:

$\sin B = - \frac{12}{13}$

Then:

$\cos 2 B = 2 {\cos}^{2} B - 1$

$\textcolor{w h i t e}{\cos 2 B} = 2 \left(1 - {\sin}^{2} B\right) - 1$

$\textcolor{w h i t e}{\cos 2 B} = 1 - 2 {\sin}^{2} B$

$\textcolor{w h i t e}{\cos 2 B} = 1 - 2 {\left(- \frac{12}{13}\right)}^{2}$

$\textcolor{w h i t e}{\cos 2 B} = 1 - \frac{288}{169}$

$\textcolor{w h i t e}{\cos 2 B} = \frac{169 - 288}{169}$

$\textcolor{w h i t e}{\cos 2 B} = - \frac{119}{169}$