Question

Знайти cos(a+b), якщо ctg a= -43 , ctg b=43

Answer 1

$ctg a=-frac43;\ left { {{sin^2 a+cos^2a=1} atop {frac{cos a}{sin a}=-frac43}}atop{sin acdotcos a<0} right.\ cos a=-frac43sin a;\ left(-frac43sin^2 aright)^2+sin^2 a=1;\ frac{16}{9}sin^2a+sin^2a=1;\ frac{25}{9}sin^2a=1; sin^2a=frac9{25};\ sin a=pmfrac35; cos a=-frac43cdotleft(pmfrac35right)=mpfrac45;$

$ctg b=frac43;\ left { {{sin^2 b+cos^2b=1} atop {frac{cos b}{sin b}=frac43}}atop{sin bcdotcos b>0} right.\ cos b=frac43sin b;\ left(frac43sin^2 bright)^2+sin^2 b=1;\ frac{16}{9}sin^2b+sin^2b=1;\ frac{25}{9}sin^2b=1; sin^2b=frac9{25};\ sin a=pmfrac35; cos a=frac43cdotleft(pmfrac35right)=pmfrac45;$тогда имеем
$cos(a+b)=cos acdotcos b-sin asin b=\ =mpfrac43 cdotpmfrac43-pmfrac43cdotpmfrac43=-frac{16}{9}-frac{16}{9}=-frac{32}9=-3frac59.$