Consider a right-angled triangle and let \(\theta\) be one of its acute angles. We define the sine of \(\theta\), written \(\sin(\theta)\), as the length of the side opposite to \(\theta\) divided by the length of the hypotenuse. Similarly, we define the cosine of \(\theta\), written \(\cos(\theta)\), as the length of the side adjacent to \(\theta\) divided by the length of the hypotenuse.
Now take a right-angled triangle with acute angle \(\alpha\), and on its hypotenuse build another right-angled triangle with acute angle \(\beta\). Use the resulting diagram to show that \(\sin(\alpha+\beta)=\sin(\alpha)\cos(\beta)+\sin(\beta)\cos(\alpha)\).