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If \( \sin \alpha=\frac{15}{53} \) and \( \sin \beta=\frac{33}{65} \), find the values of sin(α - β). |
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Answer» \(\because sin \,\alpha = \frac{15}{53}\) \(\therefore cos \,\alpha = \sqrt{1 - sin^2\alpha } \) \(= \sqrt{1 - \left(\frac{15}{53}\right)^2}\) \(=\frac{\sqrt{53^2 - 15^2}}{53}\) \(= \frac{\sqrt{2584}}{53}\) \(= \frac{2\sqrt{646}}{53}\) \(cos \beta = \sqrt{1 - sin^2\beta}\) \(= \sqrt{1 - \left(\frac{33}{65}\right)^2}\) \(= \frac{\sqrt{65^2 - 33^2}}{65}\) \(= \frac{\sqrt{3136}}{65}\) \(= \frac{56}{65}\) \(\therefore sin(\alpha - \beta) = sin\,\alpha \;cos\beta - cos\,\alpha\; sin\beta \) \(= \frac{15}{53} \times \frac{56}{65} -\frac{2\sqrt{646}}{53} \times \frac{33}{65}\) \(= \frac1{65\times 53} (840 - 66\sqrt{646})\) |
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