Question

Sabendo que sen2a = 4/5, calcule tga+cotga

Answer 1

sen²a+cos²a=1

sen2a=2*sena*cosa

2*sena*cosa=4/5 ⇒ sena*cosa=4/5*1/2 ⇒ sena*cosa=4/10

tg+cotga=tga+1/tga=(tg²a+1)/tga=[(sen²a/cos²a)+1]/[sena/cosa]=[(sen²a+cos²a)/cos²a]*[cosa/sena]=(1/cos²a)*(cosa/sena)=1/cosa*sena=1/(4/10)=1*10/4=10/4

Answer 2

 Olá!

De acordo com o enunciado,

$\\ \mathsf{\sin (2a) = \frac{4}{5}} \\\\ \mathsf{2 \cdot \sin a \cdot \cos a = \frac{4}{5}} \\\\ \maths{\sin a \cdot \cos a = \frac{2}{5}}$

 Com efeito,

$\\ \mathsf{\tan a + \cot a =} \\\\ \mathsf{\frac{\sin a}{\cos a} + \frac{\cos a}{\sin a} =} \\\\ \mathsf{\frac{\overbrace{\sin^2 a + \cos^2 a}^{1}}{\sin a \cdot \cos a} =} \\\\ \mathsf{\frac{1}{\sin a \cdot \cos a} =} \\\\ \mathsf{\frac{1}{\frac{2}{5}} =} \\\\ \mathsf{1 \cdot \frac{5}{2} =} \\\\ \boxed{\boxed{\mathsf{\frac{5}{2}}}}$