Question

Se sen(x) = -3/5 , com x pertencente ao 3º quadrante do círculo trigonométrico,
determine:
a) cos(x) b) tg(x) c) sec(x) d) cossec(x) e) cotg(x)

Answer 1

Resposta:

$cscx=-\frac{5}{3} , cosx=-\frac{4}{5} , secx=-\frac{5}{4}, tanx=\frac{3}{4}, cotx=\frac{4}{3}$

Explicação passo a passo:

We know that we have a three fundamental identities.

\$i). sin^{2} x+cos^{2} x=1\\\\ii). 1+tan^{2} x=sec^{2} x\\\\iii). 1+cot^{2} x=csc^{2} x$

From 1st fundamental identity,

$sin^{2} x+cos^{2} x=1\\\\(-\frac{3}{5} )^{2} +cos^{2} x=1\\\\ \frac{9}{25} +cos^{2} x=1\\\\cos^{2} x=1-\frac{9}{25} \\\\cos^{2} x=\frac{25-9}{25} \\\\cos^{2} x=\frac{16}{25}$

Taking square root on both sides, and we get

$cosx=-\frac{4}{5}$            ∴ x is in 3rd quadrant, cos (x) must be negative

We know that sin (x) and csc (x) are reciprocal to each other. If,  

$sinx=-\frac{3}{5}$     then,    $cscx=-\frac{5}{3}$

$cosx=-\frac{4}{5}$         Then,   $secx=-\frac{5}{4}$

$tanx=\frac{sinx}{cosx}\\\\tanx=\frac{-\frac{3}{5} }{-\frac{4}{5} } \\\\tanx = \frac{3}{4}$        ∴ x is in 3rd quadrant, tan (x) must be positive

tan(x) and cot(x) are reciprocal to each other.

$cotx=\frac{4}{3}$