How do I use trigonometric ratios to work out lengths in right-angled triangles?

There are three trigonometric ratios: sine, cosine and tangent, shortened to sin, cos and tan. Firstly, let's draw a triangle and label the sides. In a right-angled triangle, we call the bottom side between the right angle and the other known angle the adjacent side; the other side coming off the right angle is the opposite side; and the longer, third side is the hypotenuse. The key rules to remember when using the trigonometric ratios are that sin(angle)=opposite/hypotenuse; cos(angle)=adjacent/hypotenuse; and tan(angle)=opposite/adjacent. A handy acronym to remember the rules is SOH-CAH-TOA. Using these rules, together with the Pythagoras theorem, you can work out any missing side so long as you know the angle and one of the side lengths.For example, if we have a right-angled triangled, where we know one of the other angles to be 50 degrees, and the side opposite to this to be 5cm, we can use the tan rule to get adjacent=5/tan(50), which works out as 4.20, to 3 significant figures.