Question

Find the exact value of$\sin \left(1^o\right)+\sin \left(2^o\right)+\sin \left(3^o\right)+....+\sin \left({360}^o\right)$

Short answer

The exact value of$\sin \left(1^o\right)+\sin \left(2^o\right)+....+\sin \left({360}^o\right)$is$0$

Step-by-step solution

given information

We are given$\sin \left(1^o\right)+\sin \left(2^o\right)+....+\sin \left({360}^o\right)$

Use periodicity propertysinθ=sinθ-2π

We get

$$\begin{gathered} \sin \left(1^o\right)+\sin \left(2^o\right)+....+\sin \left({360}^o\right) \\ \sin \left(-{359}^o\right)+\sin \left(-{358}^o\right)+....+\sin \left({360}^o\right) \\ -\sin \left({359}^o\right)-\sin \left({358}^o\right)-....+\sin \left({360}^o\right) \\ =0 \end{gathered}$$

Conclusion

The exact value of$\sin \left(1^o\right)+\sin \left(2^o\right)+....+\sin \left({360}^o\right)is0.$