Pythagoras Theorem. Recall. Pythagoras theorem gives the relationship between the lengths of side in right angled triangles.

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1 Pythagoras Theorem Recall Pythagoras theorem gives the relationship between the lengths of side in right angled triangles. It is generally conveyed as follows: The square of the hypotenuse is equal to the sum of the square of the other two sides. In the form of an equation, this is: So what is the Hypotenuse? c 2 = a 2 + b 2 Put simply the hypotenuse is the longest side of the right-angled triangle & is the side opposite the right angle. Finding the Hypotenuse 1) Label the hypotenuse (c) and the other sides (a) and (b). 2) Use the formula above, substituting the given numbers for the letters Example Label the sides (here done for you) c 2 = a 2 + b 2 c 2 = c 2 = c 2 = 202 Now take the SQUARE ROOT to get c. c = 202 c = 14.2 cm

2 SIN, COS and TAN Recall Like Pythagoras theorem these rules only work with right angled triangles. Generally in maths problems set in exams the fact that you have a right angled triangle is not obvious it may be a case of adding a line or two based on the information you have to make one! So what are the first steps I need to do? Before tacking any problem you must work out which of the trigonometric ratios (sin, cos or tan) that you need. To do this you must label the sides of your right angled triangle correctly! This is achieved as follows: 1) Hypotenuse (H) The longest side and always opposite the right-angle 2) Opposite (O) The side directly opposite the angle you have been given / asked to find 3) Adjacent (A) The remaining side! i.e. Note that θ is the Greek letter Theta and is the common letter for unknown angles, just as x is used often for unknown lengths. Your teacher may use different letters or Greek letters, but remember it doesn t change your workings! OK, so how do I go about solving a trigonometry problem? Just follow this methodology: 1) Label your right angled triangle (see above) 2) Add the information you have been given (i.e. lengths, angles) 3) See what information you don t have and decide whether sin, cos or tan.

3 So how do I know which ratio (sin, cos, tan) to use? It is easy to remember the following mnemonic: SOH CAH TOA Hopefully you can see the letters you have been using (S = sin, C = cos, O = opposite etc) Now realising these, you can just substitute in the values you know and re-arrange the equation to find the unknown value. Example Find the unknown length x of the following right angled triangle. X cm Label the sides O = X cm 50 o 50 o 12 cm A = 12 cm H Having labelled the sides we see we have the angle and the adjacent. Using the SOH, CAH, TOA mnemonic above, we see the one which contains A (our known value) and O (unknown) is TOA, so we should use TAN as our function. Therefore: tan θ = O A tan 50 = O 12 O = tan O = 14.3 cm

4 3D Trigonometry Sounds horrible doesn t it? BUT rhis is just the same as your bog-standard, flat, 2D trigonometry. Therefore we will be applying Pythagoras and Sin, Cos or Tan. The catch is its harder to see the tight angled triangles! However once you do spot them, just do this: 1) Redraw them Flat 2) Label the sides 3) Add the information you know 4) Work out what you want/need as before Example The diagram below shows a wedge of the world largest cheddar cheese in which the rectangle PQRS is perpendicular (at 90 o to) rectangle RSTU. The distances are shown on the diagram. Calculate: (a) the distance QT (b) The angle QTR Answer (a) Well the first thing we need to work out is what we are actually trying to work out! We need the line QT: So let s draw it in Now, recall we need to look for the right angled triangles. Look at TQR that make a right angled triangle. And even better we know how long QR is, so we just need to work out the length TR.

5 Let s find TR: OK looking carefully, we can see a right angled triangle on the base of the cheese wedge. So let us redraw that triangle (TRU) flat. So we have two sides and want the hypotenuse, so we use Pythagoras: c 2 = c 2 = c 2 = c = c = 9.21 m Now all we need to do is calculate TQ: We need to identify and draw a right angled triangle again: Now again we apply Pythagoras on this new triangle, and we get c = 9.54 m.

6 (b) We now need to find the angle QTR. So let s update our diagram with what we know from part (a). We can now draw our right angled triangle: So again now a choice of Sin, Cos or Tan Here we actually know all sides, so we can use any other the identities, so let us use Cos this time: cos θ = a h cos θ = cos θ = Now we need to Use inverse cosine (or invers sin or tan if we used those ratios) on our calculators cos 1 θ = θ = 15.2 o