Answer to Question #161491 in Trigonometry for Ella

Question

If cos(x) = -(5/13) and π/2 ≤ x ≤ π, find the value of sin(x) and tan(x).

Answer 1

Solution: Given that cos(x) = -($\frac{5}{13}$) = $\frac{-5}{13}$ and $\pi /2 \leq x \leq \pi$

$\therefore$ x is in second quadrant.

{ we know that if cos(x) = $\frac{b}{h}$ then sin(x) = $\frac{a}{h}$ , where $a$ = $\sqrt{h^2 - b^2}$ & tan(x) = $\frac{sin(x)}{cos(x)}$ }

therefore if we compare with the given cos(x) value we get:

b = -5 & h = 13

$\implies$ $a$ = $\sqrt{13^2 - (-5)^2}$

$\implies$ $a$ = $\sqrt{169 - 25}$

$\implies$ $a$ = $\sqrt{144}$

$\implies$ $a$ = $\pm$ 12

in the second quadrant value of sin(x) is always positive.

therefore, we take $a$ = 12

sin(x) = $\frac{a}{h}$ = $\frac{12}{13}$

tan(x) = $\frac{sin(x)}{cos(x)}$ = $\frac{12/13}{-5/13}$ = -$\frac{12}{5}$