SSCE HSC Mathematics Extension 2 Forces and equations of motion: In $ \triangle ABC $, \( \angl

Question

In $ \triangle ABC $, $ \angle CAB = \alpha, \angle ABC = \beta $ and $ \angle BCA = \gamma $. The point $ O $ is chosen inside $ \triangle ABC $ so that $ \angle OAB = \angle OBC = \angle OCA = \theta $, as shown in the diagram.

(i) Show that $ \frac{OA}{OB} = \frac{\sin(\beta - \theta)}{\sin(\theta)} $

(ii) Hence show that $ \sin^{3}(\theta) = \sin(\alpha - \theta) \sin(\beta - \theta) \sin(\gamma - \theta) $

(iii) Prove the identity $ \cot x - \cot y = \frac{\sin(y - x)}{\sin x \sin y} $

(iv) Hence show that $ (\cot \theta - \cot \alpha)(\cot \theta - \cot \beta)(\cot \theta - \cot \gamma) = \csc \alpha \csc \beta \csc \gamma $

(v) Hence find the value of $ \theta $ when $ \triangle ABC $ is an isosceles right triangle.

In \( \triangle ABC \), \( \angle CAB = \alpha, \angle ABC = \beta \) and \( \angle BCA = \gamma \) - HSC - SSCE Mathematics Extension 2 - Question 6 - 2006 - Paper 1

Worked Solution & Example Answer:In \( \triangle ABC \), \( \angle CAB = \alpha, \angle ABC = \beta \) and \( \angle BCA = \gamma \) - HSC - SSCE Mathematics Extension 2 - Question 6 - 2006 - Paper 1

In an alien universe, the gravitational attraction between two bodies is proportional to \( x^{-3} \), where \( x \) is the distance between their centres.

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