C 2 Trigonometry KUS objectives BAT know the

C 2: Trigonometry • KUS objectives BAT know the key features of trig graphs using both degrees and radians BAT use exact values of trig functions to solve problems BAT recognise and sketch transformations to trig graphs Starter: calculate What patterns can you see?

WB 10 Trig graphs https: //www. geogebra. org/m/S 2 g. Mrkb. D y y = sinθ 1 You should know these intimately! -360º -270º -180º -90º 0 90º 180º 270º θ 360º -1 Describe the features of each to your neighbour - Similarities y 1 -360º -270º -180º -90º 0 90º 180º 270º θ 360º -1 - Intersections with the axes y = tanθ - Period of each graph - Asymptotes on the tan graph y = cosθ 1 -360º -270º -180º -90º 0 -1 90º 180º 270º θ 360º

WB 11 a Symmetry of sin graph You need to be able to recognise the graphs of sinθ, cosθ and tanθ You need to be able to work out larger values of sin, cos and tan as acute angles (0º - 90º) y = sinθ y 1 -40 50 0 130 θ 90º 180º 270º 360º -1 Write sin 130º as sine of an acute angle (sometimes asked as a ‘trigonometric ratio’) Sin 130º = Sin 50º Draw a sketch of the graph Mark on 130º Using the fact that the graph has symmetry, find an acute value of θ which has the same value as sin 130

WB 11 b Symmetry of cos graph y = cosθ y You need to be able to recognise the graphs of sinθ, cosθ and tanθ You need to be able to work out larger values of sin, cos and tan as acute angles (0º - 90º) +60 -60 +30 -270º -180º +30 -90º -120 1 +60 60 0 θ 90º 180º 270º -1 Write cos (-120)º as cos of an acute angle Cos(-120)º = -Cos 60º Draw a sketch of the graph Mark on -120º Using the fact that the graph has symmetry, find an acute value of θ which has the same numerical value as cos (-120) The value you find here will have the same digits in it, but will be multiplied by -1

WB 11 c Symmetry of tan graph y = tanθ 1 0 -1 θ

WB 11 d More Symmetry of trig graphs y y = sinθ 1 For any angle ϴ, -360º -270º -180º -90º 0 90º 180º 270º θ 360º -1 sin(- ϴ) = - sinϴ y cos(- ϴ) = cosϴ y = cosθ 1 tan(- ϴ) = - tanϴ -360º -270º -180º -90º 0 90º 180º 270º θ 360º -1 Complete: y = tanθ sin(- 25) = cos(-240) = tan(- 110) = 1 -360º -270º -180º -90º 0 -1 90º 180º 270º θ 360º

6 A

WB 12 a Exact values 1 Some values of Sin, Cos or Tan can be written using fractions, surds, or combinations of both… 2 60˚ We can use an Equilateral Triangle with sides of length 2 to show this. Using Pythagoras, the missing side in the right angled triangle is √ 3 (Square root of 22 -12) Work out sin, cos and tan of the angles shown 60˚ 2 Hyp 2 30˚ 60˚ Opp 1 √ 3 Opp

WB 12 b Exact values 2 Opp Sinθ = Hyp Sin 30 = Sin 60 = 1 2 60˚ 2 √ 3 2 60˚ 2 Adj Cosθ = Hyp √ 3 Cos 30 = 2 1 Cos 60 = 2 Opp Tanθ = Adj 1 √ 3 Tan 30 = √ 3 = 3 Tan 60 = √ 3 Hyp 2 30˚ 60˚ Opp 1 √ 3 Opp

WB 12 c Exact values 3 We can also do a similar demonstration with a right-angled Isosceles triangle, with the equal sides being of length 1 unit. Using Pythagoras’ Theorem, the hypotenuse will be of length √ 2 (Square root of 12 + 12) Hyp Opp 1 √ 2 45˚ 1 1 √ 2 Sin 45 = √ 2 = 2 1 √ 2 Cos 45 = √ 2 = 2 Tan 45 = 1 1 = 1

WB 13 Exact values proofs

WB 14 Exact values Hyp 5 3 4 Opp Opp

KUS objectives BAT know the key features of trig graphs using both degrees and radians BAT use exact values of trig functions to solve problems self-assess One thing learned is – One thing to improve is –

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