Question

(Fuvest 1994) O valor de (tg 10° + cotg 10°)sen 20° é:

Answer 1

Seno do arco duplo $\mathsf{\Longrightarrow}$

$\mathsf{sen(2\cdot \alpha) \ = \ 2 \cdot sen(\alpha) \cdot cos(\alpha)}$

$\mathsf{tg(\alpha) \ = \ \dfrac{sen(\alpha)}{cos(\alpha)} \ \therefore \ \underbrace{\mathsf{cotg(\alpha)}}_{tg^{-1}(\alpha)} \ = \ \dfrac{cos(\alpha)}{sen(\alpha)}}}$

Seja $\mathsf{\alpha \ = \ 10^\circ \ \longrightarrow}$

$\mathsf{(tg(10^\circ) \ + \ cotg(10^\circ)) \ \cdot sen(20^\circ) \ \rightarrow} \\ \\ \\ \mathsf{(tg(\alpha) \ + \ cotg(\alpha)) \ \cdot \ sen(2 \cdot \alpha) \ \rightarrow} \\ \\ \\ \mathsf{\bigg(\dfrac{sen(\alpha)}{cos(\alpha)} \ + \ \dfrac{cos(\alpha)}{sen(\alpha)}\bigg) \ \cdot \ 2 \cdot sen(\alpha) \ \cdot \ cos(\alpha) \ \rightarrow}$

$\mathsf{\bigg(\underbrace{\mathsf{\dfrac{sen(\alpha) \cdot sen(\alpha) \ + \ cos(\alpha) \cdot cos(\alpha)}{sen(\alpha) \ \cdot \ cos(\alpha)}}}_{soma \ de \ fra\c{c}\~oes}\bigg) \ \cdot 2 \ \cdot \ sen(\alpha) \ \cdot \ cos(\alpha) \ \rightarrow} \\ \\ \\ \\ \mathsf{\bigg(\dfrac{\overbrace{\mathsf{sen^2(\alpha) \ + \ cos^2(\alpha)}}^{rela\c{c}\~ao \ fundamental \ = \ 1}}{sen(\alpha) \ \cdot \ cos(\alpha)}\bigg) \ \cdot \ 2 \ \cdot \ sen(\alpha) \ \cdot \ cos(\alpha) \ \rightarrow} \ \rightarrow$

$\mathsf{\dfrac{2 \ \cdot \ sen(\alpha) \cdot cos(\alpha)}{sen(\alpha)\ \cdot \ cos(\alpha)} \ = \ \boxed{\boxed{2}}}$

$\mathsf{\heartsuit \ \mathbb{JN}}$