[ GM4.4 Extension Plenary] Trigonometry [ GM4.4 Extension Plenary] This is part of a special spiral made from straight lines. It is named after the mathematician Theodorus. a) Find the lengths of the first few spokes. Leave the square roots () in – do not work them out. (The first spoke is 1 unit length and begins at the centre O.) b) What is the length of the 20th spoke? Calculate the angle between the i) 2nd and 3rd ii) 3rd and 4th spoke. Challenge d) After how many spokes will the spokes make half a complete turn? O Preamble A challenging activity involving Pythagoras and using the inverse tangent to find angles. Pupils may need some initial support. Possible content Using Pythagoras, pattern spotting and simple trigonometry. Resources None although, given the amount of numerical processing, some pupils may find using a spreadsheet appropriate. Solution/Notes a) Using Pythagoras gives: b) So the length of the 20th spoke is . c) i) 30° ii) 26.6° d) The 9th spoke (more capable pupils could be encouraged to use a spreadsheet for the “number crunching”.) Original Material © Cambridge University Press 2010 Original Material © Cambridge University Press 2010