Question

Sabendo que sec x = - 5/4 , x E 2° quadrante , então o valor de y = cos x + sen x , é:

a - 1/5
b) - 7/5
c) 1/5
d) 3/5
e) 7/5​

Answer 1

Resposta: y = - 1/5 — Letra a)

Explicação passo-a-passo:

sec(x) = - 5/4 =>

1/cos(x) = - 5/4 =>

cos(x) = - 4/5

Sabe-se que cos(x) = - 4/5, então cos²(x) = 16/25. Logo:

cos²(x) = 16/25 e sen²(x) + cos²(x) = 1 = 25/25 =>

16/25 + sen²(x) = 25/25 =>

sen²(x) = 25/25 - 16/25 =>

sen²(x) = 9/25 =>

sen²(x) = 3²/5² e pi/2 < x < pi =>

|sen(x)| = sen(x) = 3/5

Assim sendo, o valor de y será:

y = cos(x) + sen(x) =>

y = - 4/5 + 3/5 =>

y = - 1/5

Abraços!

Answer 2

Resposta:

A)

Explicação passo-a-passo:

$sec x = \frac{1}{cos x} \\\\ sen^2 x + cos ^2 x = 1 \\\ \frac{sen^2 x}{cos^2 x} + \frac{cos^2 x}{cos^2 x} = \frac{1}{cos^2 x} \\\\ tg^2 x + 1 = sec^2 x \\\ tg^2 x + 1 = (- \frac{5}{4})^2 \\\\ tg^2 x = \frac{25}{16} - 1 \\\\ tg^2 x = \frac{9}{16} \\\\ tg x = \pm \sqrt{\frac{9}{16}} \\\\ tg x = \pm \frac{3}{4} \\\\ x E quadrante 2 \\\ tg x = - \frac{3}{4} \\\\ tg x = \frac{sen x}{cos x} \\\\ \frac{y}{cos x} = \frac{cos x}{cos x} + \frac{sen x}{cos x} \\\\ \frac{y}{cos x} = 1 + tg x \\\\ 1 - \frac{3}{4} = \frac{y}{cos x} \\\\ \frac{y}{cos x} = \frac{1}{4} \\\\ y.\frac{1}{cos x} = \frac{1}{4} \\\\ y.sec x = \frac{1}{4} \\\\ -\frac{5}{4}y = \frac{1}{4} \\\\\ y = - \frac{\frac{1}{4}}{\frac{5}{4}} \\\\\\ y = - \frac{1}{5}$

Espero ter ajudado.