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The value of

Question:

The value of $\frac{\sin 70^{\circ}}{\sin 110^{\circ}}$ is ____________ .

Solution:

$\frac{\sin 70^{\circ}}{\sin 110^{\circ}}=\frac{\sin 70^{\circ}}{\sin \left(90^{\circ}+20^{\circ}\right)}$

$=\frac{\sin 70^{\circ}}{\cos 20^{\circ}} \quad\left(\right.$ Since $\left.\sin \left(90^{\circ}+\theta\right)=\cos \theta\right)$

$=\frac{\sin 70^{\circ}}{\cos \left(90^{\circ}-70^{\circ}\right)}$

$=\frac{\sin 70^{\circ}}{\sin 70^{\circ}}=1 \quad\left(\because \cos \left(90^{\circ}-\theta\right)=\sin \theta\right)$

Hence $\frac{\sin 70^{\circ}}{\sin 110^{\circ}}=1$

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