Mr Barton’s Maths Notes Trigonometry 1. Pythagoras www.mrbartonmaths.com

1. Pythagoras What is Pythagoras’ Theorem? Pythagoras’ Theorem is probably the most famous theorem in the history of mathematics It was “invented” by a Greek named Pythagoras (or one of his loyal followers who always marked any of their discoveries with the Pythagoras brand) somewhere around 6BC Pythagoras discovered a very important relationship between the lengths of sides in a right-angled triangles: “If you take the lengths of the two shortest sides of any right-angled triangle, square them and add the answers together, you end up with the square of the longest side (the hypotenuse)”

3. The two forms of Pythagoras’ Theorem 2. What is the Hypotenuse? In order to use Pythagoras’ Theorem (or all the trig that is coming around the corner!), you must be an expert at finding the Hypotenuse of any right angled triangle The Hypotenuse is the longest side of the right-angled triangle, and it is the side opposite the right-angle! hypotenuse hypotenuse hypotenuse 3. The two forms of Pythagoras’ Theorem Pythagoras’ Theorem can be written in two ways depending on whether you want to find the length of the hypotenuse of a triangle, or one of the other sides. The two ways are just different arrangements of the same original formula, so if you are good at formula re-arranging, then you only need to remember one!

c a b 4. Finding the Hypotenuse 1. Label the Hypotenuse c, and the other sides a and b 2. Use the following formulae: 3. Replace the letters with the numbers you have been given, and carefully do the sum! c a b

c a b 5. Finding a side that isn’t the Hypotenuse 1. Label the Hypotenuse c, label the side you want to find a, and the other side b 2. Use the following formulae: 3. Replace the letters with the numbers you have been given, and carefully do the sum! Note: As I mentioned before, this version of the formula is just a different arrangement of: Just subtract b2 from both sides and you should see what I mean! c a b

Examples 1. Okay, so the side we want to find is the Hypotenuse, so let’s go through our routine: 1. Label the sides 2. Use the formula: 3. Put in the numbers: c a ? 9 cm 11 cm b Square root both sides! Note: Our answer is longer than both our other sides… which is good because the hypotenuse is supposed to be the longest side!

2. Okay, so the side we want to find is NOT the Hypotenuse, so let’s go through our routine: 1. Label the sides 2. Use the formula: 3. Put in the numbers: ? a 10.2 m c 3.1 m Square root both sides! b Note: Our answer is shorter than our hypotenuse… which is good because the hypotenuse is supposed to be the longest side!

3. A 5m ladder rests against the side of a house. The foot of the ladder is 1.5m away from the house. How far up the side of the house does the ladder reach? At first glance this question does not appear to have anything to do with Pythagoras, but in these sort of situations, always follow this advice: IF IT’S TRICKY, DRAW A PICCY!!!… and then look what we have! It’s just a right angled triangle and we want to find a side that is NOT the Hypotenuse, so let’s go through our routine: 1. Label the sides 2. Use the formula: 3. Put in the numbers: c 5 m ? a 1.5 m b Square root both sides!

4. Find the distance between these two co-ordinates: (4, 5) and (-2, 1) Again, at first glance this question does not appear to have anything to do with Pythagoras, but if we do a quick sketch of our co-ordinates, then look what we have! (4, 5) It’s just a right angled triangle and we want to find the Hypotenuse, so let’s go through our routine: 1. Label the sides 2. Use the formula: 3. Put in the numbers: c a 4 (-2, 1) b 7 To work out the lengths of the sides, we just count how many squares would be in between on a co-ordinate grid! Square root both sides!

Good luck with your revision!