Hello, r_maths!
The diagram shows a kite whose leading diagonal forms the diameter of a circle.
Find the exact value of $\displaystyle \sin(XOY)$.
Look at half of the kite . . . Code:
Z
*
| * _
| *√5
| *
__ | 90° * Y
√14 | *
| *
| *
| * 3
|θ *
| *
|*
*
O
Since $\displaystyle ZO$ is a diameter, $\displaystyle \angle Y = 90^o.$
Using Pythagorus, we find that: .$\displaystyle ZO = \sqrt{14}$
Let $\displaystyle \theta = \angle ZOY$
Then: .$\displaystyle \sin\theta = \frac{\sqrt{5}}{\sqrt{14}},\;\cos\theta = \frac{3}{\sqrt{14}} $
Then: .$\displaystyle \sin(XOY) \:=\:\sin(2\theta) \:=\:2\sin\theta\cos\theta \:=\:2\left(\frac{\sqrt{5}}{\sqrt{14}}\right)\left (\frac{3}{\sqrt{14}}\right) \:=\:\frac{3\sqrt{5}}{7} $