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| Sine Formula (Sine Rule) |
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| Prove that in any triangle, the sides of a triangle are proportional to the sines of the opposite angles. |
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| Prove that each side of a triangle is proportional to sine of the angle between the other two i.e., |
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| Proof : |
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| Let ABC be any triangle. |
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| Draw AD ^ BC. (Produce BC if necessary in (ii)) |
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| From (1) and (2), we get |
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| DABC is an obtuse-angled triangle. |
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| From (4) and (5), we get |
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| DABC is a right-angled triangle at C. |
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| Similarly, by drawing perpendicular from B to AC, we can prove that |
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| Hence from (6) and (9), we get |
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| Example: |
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| where R = Circumradius of DABC. |
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| Suggested answer: |
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| Fig (iii) |
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| Consider triangles (acute, obtuse and right angle) as shown in figs (i), (ii) and (iii). |
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| Let O be the centre of the circumscribed DABC. |
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| Let BD = 2R be the diameter of the circle. Draw CD, then we have |
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| From the first figure, we have |
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| From the second figure , we have |
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| From the third figure , we have |
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