Question

Знайти cos(a+b), якщо ctg a= -4\3 , ctg b=4\3

Answer 1

$ctg a=-\frac43;\\ \left \{ {{\sin^2 a+\cos^2a=1} \atop {\frac{\cos a}{\sin a}=-\frac43}}\atop{\sin a\cdot\cos a$

$ctg b=\frac43;\\ \left \{ {{\sin^2 b+\cos^2b=1} \atop {\frac{\cos b}{\sin b}=\frac43}}\atop{\sin b\cdot\cos b0} \right.\\ \cos b=\frac43\sin b;\\ \left(\frac43\sin^2 b\right)^2+\sin^2 b=1;\\ \frac{16}{9}\sin^2b+\sin^2b=1;\\ \frac{25}{9}\sin^2b=1;\ \ \sin^2b=\frac9{25};\\ \sin a=\pm\frac35;\ \ \cos a=\frac43\cdot\left(\pm\frac35\right)=\pm\frac45;\\$тогда имеем
$\cos(a+b)=\cos a\cdot\cos b-\sin a\sin b=\\ =\mp\frac43 \cdot\pm\frac43-\pm\frac43\cdot\pm\frac43=-\frac{16}{9}-\frac{16}{9}=-\frac{32}9=-3\frac59.$