As alarm bells ring, fire-fighters spring into action, drop down their pole and head for the fire engine. In no time they have safely set a ladder against a burning building and are in position to perform a rescue. Ben appears, and explains that this is a training exercise, but that in all their work the fire brigade must get the angles of their ladders just right.

00.46—02.02

Katie demonstrates that the ground, ladder and building form a right-angled triangle, and she identifies the angle that is crucial to safety. Ben explains how in practice the fire-fighters estimate the correct angle, using experience and a ‘rule of thumb’. He measures this angle with a protractor. Katie shows how her diagram is in the same proportions as the real situation, and so the angle of her ladder is the same as Ben’s.

02.02—03.21

In the studio, Katie revises the terminology of right-angled triangles. She shows how to identify and name the sides in relation to an angle. We learn that the ratio of the opposite side to the hypotenuse is constant for a given angle in any right-angled triangle.

03.21—04.11

Returning to the ladder, Ben explains that the height it reaches is the ‘opposite’ side in this case, and the ‘hypotenuse’ is the length of the ladder. He finds the value of the sine ratio for these lengths. Katie calculates the same ratio from her scale diagram and gets the same answer as Ben, pointing out that this is because her angle is the same.

04.11—05.56

The sine ratio is defined, and the formula ‘sin = opposite / hypotenuse’ given. Katie works through a typical example, showing us how to find the height that a ladder at a given angle can reach up a wall.

05.56—06.45

Rachel Crosland gives a stunning display of her skill as a water-skier, including the jumps from ramps that she performs in competitions. She explains to Ben that for competition jumps the take-off height and the angle of the ramp are always the same.

06.45—08.15

Ben uses Rachel’s information to calculate the length of the ramp — using the sine formula to find the hypotenuse. If only water-skiing were as straightforward! After a few dodgy attempts, Ben takes off behind the boat.

08.15—11.37

In ‘Tick or Trash’, Ben and Katie tackle a problem that involves using the sine formula to calculate a side in a right-angled triangle. Ben’s mistake allows him to look in more detail at how to identify the sides correctly before substitution.

11.37—12.53

Tanni Grey competes in many different events in the Paralympics. She explains that ramps are helpful for wheelchair users, but only if they are built at an angle that makes them easy for everyone to use. Tanni tests three ramps at 15, 10 and 5 degrees. Even she can only just manage the steepest, commenting that descending it is quite dangerous. The 10 degrees ramp is still hard work; but at 5 degrees the ramp is comfortable and Tanni recommends this angle to anyone thinking of building an access ramp.

12.53—end

Katie measures another ramp, finding its length and the height at the kerb. She explains how these values and the sine formula can be used to find the angle, and shows us the inverse-sine function on her calculator. She leaves the viewers to work out whether the ramp shown would be suitable for wheelchair users.