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Trigonometry Objectives of this slideshow Identify opposite, hypotenuse and adjacent sides Define SOCAHTOA Apply SOCAHTOA depending on which rule we need Use all of the above to help find missing angles in a right angled triangle Calculate missing sides of a right angled triangle given another angle and length of one side.

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opposite Hypotenuse adjacent Hypotenuse adjacent opposite Hypotenuse adjacent Hypotenuse opposite Hypotenuse opposite adjacent opposite

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C B A Label the opposite, adjacent and hypotenuse A = adjacent B = opposite C = hypotenuse Top tips always find: Hypotenuse first as this is the easiest (always opposite the right angle) Opposite second as it is always opposite the angle we are interested in Finally the adjacent (which means next to) this one is the only one left and is next to the angle but is not the hypotenuse

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C B A A = adjacent B = opposite C = hypotenuse A = opposite B = adjacent C = hypotenuse Label the opposite, adjacent, and hypotenuse here. If you have any questions write them on your whiteboard Does the angle we are looking at make a difference to the names of the sides? YES IT DOES!!!!

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You need to remember this Sine = opposite hypotenuse Cosine = adjacent hypotenuse Tangent = opposite adjacent

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The two sides are equal length so it must be an isosceles triangle. It can’t be an equilateral as one of the angles is a right angle. 180° – 90° = 90° 90° ÷ 2 = 45° The angle must be 45° 6cm You should be able to tell what the size of this angle is using your knowledge of triangles. We are going to prove this using trigonometry. The reason I have picked an angle that we already know is so that we can be sure we are getting the correct answer using trigonometry which we are unfamiliar with. Once we have become familiar with it we can start looking at more difficult examples that we can only solve using trigonometry. Step 1: label the sides H, O and A (this order) Step 2: work out which trigonometric function we need to use based on the info we are given. In this case we are given the O and A which relates to tangent because Tangent = O/A Step 3: the tangent of this angle = 6/6 =1 to find the actual angle we do tan¯¹ (1) = 45°

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13cm 5cm Have a go at this question to see if you can find the yellow angle? Click to the next screen for a little help Step 1: label the sides H, O and A Step 2: which function do you use? SOHCAHTOA Step 3: use the sine rule because sine = opp/hyp and these are the two we know. Opp/hyp = 5/13 Sine¯¹ = Tip if the answer you get seems like a wrong answer check through the steps that you have labelled the sides correctly, used the correct function and if not you probably didn’t press the inverse function. You probably pressed Sine and not Sine¯¹.

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Cosine¯¹(13/9) = ____ why??? Because cosine is adjacent/hypotenuse and the hypotenuse is always the biggest side. In this case the adjacent side was bigger than the hypotenuse therefore impossible to do. Don’t shout out your answer!!!

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4cm 3cm 8cm 3cm 6cm 5cm 7cm 4cm Sine (O/H) Cosine (A/H) tangent (O/A) Sine (O/H) Draw these triangles in your book (not to scale) and then work out the missing angle sine¯¹(3/4) = 48.6° (1dp) cosine¯¹(3/8) = 67.98° (2dp) tangent¯¹(6/5) = 50.2° (1dp) sine¯¹(4/7) = 34.8° (1dp)

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15cm 11cm Find all the missing angles

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4cm 3cm 8cm 3cm 6cm 5cm 7cm 4cm Sine (O/H) Cosine (A/H) tangent (O/A) Sine (O/H) Draw these triangles in your book (not to scale) and then work out the missing angle sine¯¹(3/4) = 48.6° (1dp) cosine¯¹(3/8) = 67.98° (2dp) tangent¯¹(6/5) = 50.2° (1dp) sine¯¹(4/7) = 34.8° (1dp)

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45° x Find the length of the side labelled x 15cm Step one what rule do we need to use? Sine because we know the hypotenuse and we want to know the opposite This time we can say that sine(45 °) = X 15cm We can multiply both sides By 15 to get 15 x sine(45 °) = X 10.61cm(2dp) = X

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45° 8cm Find the length of the side labelled x X Step one what rule do we need to use? Cosine because we need to know the hypotenuse and we do know the adjacent This time we can say that Cosine(45 °) = 8cm X We can multiply both sides By X to get X Cosine(45 °) = 8cm Then we can divide both sides by Cosine (45 °) X = 8cm ÷ Cosine(45 °) X = 11.31cm (2dp)

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4cm x x 3cm 6cm X X 4cm Draw these triangles in your book (not to scale) and then work out the missing angle 45° 60° 52° 36°

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45° Show the angle is 45° on any right angled isosceles triangle using trigonometry

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