Kason Murray

2023-01-01

What is evaluate $\mathrm{sin}\left(-t\right)$ and $\mathrm{csc}\left(-t\right)$ if $\mathrm{sin}\left(t\right)=-\frac{2}{5}?$

Gremoli9bs

Expert

Recall that $\mathrm{sin}\left(x\right)$ is an odd function, meaning that $\mathrm{sin}\left(-x\right)=-\mathrm{sin}\left(x\right)$.
So, this means that $\mathrm{sin}\left(-t\right)=-\mathrm{sin}\left(t\right)$
Knowing that already $\mathrm{sin}\left(t\right)=-\frac{2}{5}$, so $\mathrm{sin}\left(-t\right)=-\mathrm{sin}\left(t\right)=-\left(-\frac{2}{5}\right)=\frac{2}{5}$.
$\mathrm{csc}\left(t\right)=\frac{1}{\mathrm{sin}\left(t\right)}$
$\mathrm{csc}\left(-t\right)=\frac{1}{\mathrm{sin}\left(-t\right)}$
Then we calculated $\mathrm{sin}\left(-t\right)$:
$\mathrm{csc}\left(-t\right)=\frac{1}{\frac{2}{5}}=\frac{5}{2}$ (Remember, $\frac{1}{\frac{a}{b}}=\frac{b}{a}$).

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