Question

Find the exact value of each expression. Do not use a calculator.

$\sin \left({310}^{°}\right)csc\left(-{50}^{°}\right)$.

Short answer

The exact value for the expression$\sin \left({310}^{°}\right)csc\left(-{50}^{°}\right)$is,$1$.

Step-by-step solution

Step 1 Given expression is,

$\sin \left({310}^{°}\right)csc\left(-{50}^{°}\right)$.

The objective is, we need to find the exact value for the given expression without using the calculator.

Step 2 we know that,

$$\begin{gathered} \sin \left({360}^{°}-\theta \right)=-\sin \theta \\ csc\theta =\frac{1}{\sin \theta } \end{gathered}$$

So, the given expression becomes,

$$\begin{gathered} \sin \left({310}^{°}\right)csc\left(-{50}^{°}\right)=\sin \left({310}^{°}-{50}^{°}\right)csc\left(-{50}^{°}\right) \\ =-\sin {50}^{°}\times -csc\left({50}^{°}\right) \\ =-\sin {50}^{°}\times -\frac{1}{\sin {50}^{°}} \\ =1 \end{gathered}$$