I am studying A level maths on my own as an interest. I have the following problem:
In $\triangle ABC, AB = 9 cm, AC = 12 cm, \angle B = 2\theta, \angle C = \theta.$
Without using tables or calculators find $\cos\theta$ and the length of BC.
Using the cosine rule I have established that:$$\cosθ = \frac{(x^2 + 63)}{24x}$$ and $$\cos2\theta = \frac{x^2 – 63}{18x}$$ where x = BC.
I can also see that $\angle A = π – 3θ$ and that using the cosine rule $\cos 3\theta = \frac{x^2 – 225}{216}$
That’s as far as I can go.
Best Answer
Using sine rule , $$\frac{\sin {2\theta}}{12} = \frac {\sin \theta}{9}$$ $$ \frac{2\sin\theta\cos\theta}{\sin\theta} = \frac{12}9$$ $$\cos\theta = \frac 23.$$