Question

If A is in the $3$rd quadrant and B is in the fourth quadrant and $\cos A=-\frac{15}{17},\cos B=\frac{4}{5}$ find the value of $\cos (A+B)$

• A. $-\frac{84}{85}$
• B. $\frac{84}{85}$
• C. $-\frac{34}{85}$
• D. None of these

Answer 1

The correct option is A $-\frac{84}{85}$

$\cos A=-\frac{15}{17}$
$\therefore \sin A=\sqrt{1-{\cos }^{2}A}$
$=-\sqrt{1-\frac{225}{289}}=-\frac{8}{17}$
$\cos B=\frac{4}{5}$
$\therefore \sin B=-\sqrt{1-{\cos }^{2}A}$
$=-\sqrt{1-\frac{16}{25}}=-\frac{3}{5}$
Now $\cos (A+B)=\cos A\cos B-\sin A\sin B$
$=-\frac{15}{17}\times \frac{4}{5}-\frac{-8}{17}\times \frac{-3}{5}$
$=\frac{-60}{85}-\frac{24}{85}=-\frac{84}{85}$