Question

ntr-un triunghi dreptunghic ABC(M(Â)=90 de grade:
A) daca tangentaB+ ctgB=5 sa se calculeze:
1) sinB + cos B
2)tgB la puterea 2 +ctgBla puterea 2

Answer 1

Salut,

$tgB+ctgB=\dfrac{sinB}{cosB}+\dfrac{cosB}{sinB}=\dfrac{sin^2B+cos^2B}{sinB\cdot cosB}=\dfrac{1}{sinB\cdot cosB}\Rightarrow\\\\\Rightarrow sinB\cdot cosB=\dfrac{1}5;\\\\(sinB+cosB)^2=sin^2B+2\cdot sinB\cdot cosB+cos^2B=1+2\cdot sinB\cdot cosB=\\\\=1+2\cdot\dfrac{1}{5}=\dfrac{7}{5}\Rightarrow sinB+cosB=\pm\sqrt{\dfrac{7}5};\\\\(tgB+ctgB)^2=tg^2B+2\cdot tgB\cdot ctgB+ctg^2B=tg^2B+ctg^2B+2\Rightarrow\\\\\Rightarrow tg^2B+ctg^2B=(tgB+ctgB)^2-2=25-2=23.$

Green eyes.