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Question
If in ∆ABC, ∠A = 45°, ∠B = 60° and ∠C = 75°, find the ratio of its sides.
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Solution
Let \[\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = k\]
Then,
\[ \Rightarrow \frac{a}{\frac{1}{\sqrt{2}}} = \frac{b}{\frac{\sqrt{3}}{2}} = \frac{c}{\frac{1}{2\sqrt{2}}\left( 1 + \sqrt{3} \right)} \left[ \because \sin75° = \sin\left( 30°°° + 45° \right) = \sin30°\cos45° + \sin45°\cos30° \right]\]
On multiplying by \[2\sqrt{2}\]
Hence, the ratio of the sides is \[2 : \sqrt{6} : \left( 1 + \sqrt{3} \right)\]
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