Understanding Trigonometric Graphs and Patterns
Explore trigonometric graphs, key features, and transformations. Learn to calculate sine and cosine values, recognize symmetries, and work with acute angles for solving trigonometric problems. Dive into the world of sine, cosine, and tangent functions with visual aids and practice exercises.
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Trig Graphs
C2: Trigonometry 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 KUS objectives Starter: calculate sin? when ? = 10,20,30,40, ,90 What patterns can you see? cos? when ? = 10,20,30,40, ,180 ? when sin ? = 0.1,0.2,0.3,0.4, ,1
https://www.geogebra.org/m/S2gMrkbD WB 10 Trig graphs y y = sin 1 You should know these intimately! 0 -90 -360 360 -180 90 180 270 -270 -1 Describe the features of each to your neighbour y y = cos 1 0 -180 -90 -360 -270 360 90 180 270 - - Similarities Intersections with the axes Period of each graph Asymptotes on the tan graph -1 y = tan - 1 - 0 -180 -90 90 -360 -270 360 180 270 -1
WB 11a Symmetry of sin graph y = sin y You need to be able to recognise the graphs of sin , cos and tan -40 -40 1 50 130 You need to be able to work out larger values of sin, cos and tan as acute angles (0 - 90 ) 0 360 90 180 270 -1 Write sin 130 as sine of an acute angle (sometimes asked as a trigonometric ratio ) Draw a sketch of the graph Mark on 130 Using the fact that the graph has symmetry, find an acute value of which has the same value as sin 130 Sin 130 = Sin 50
WB 11b Symmetry of cos graph y = cos y You need to be able to recognise the graphs of sin , cos and tan +60 +60 1 -60 60 +30 0 +30 -90 -180 90 180 270 -270 You need to be able to work out larger values of sin, cos and tan as acute angles (0 - 90 ) -120 -1 Write cos (-120) as cos of an acute angle Draw a sketch of the graph Mark on -120 Using the fact that the graph has symmetry, find an acute value of which has the same numerical value as cos (-120) Cos(-120) = -Cos 60 The value you find here will have the same digits in it, but will be multiplied by -1
WB 11c Symmetry of tan graph y = tan You need to be able to recognise the graphs of sin , cos and tan 1 0 90 180 360 270 You need to be able to work out larger values of sin, cos and tan as acute angles (0 - 90 ) -1 Write tan 240 as tan of an acute angle Draw a sketch of the graph Mark on 240 Using the fact that the graph has symmetry, find an acute value of which has the same numerical value as tan 240 Tan 240 = Tan 60
WB 11d More Symmetry of trig graphs y y = sin 1 For any angle , 0 -90 -360 360 -180 90 180 270 -270 -1 sin(- ) = - sin y cos(- ) = cos y = cos 1 tan(- ) = - tan 0 -180 -90 -360 -270 360 90 180 270 -1 Complete: y = tan sin(- 25) = 1 0 cos(-240) = -180 -90 90 -360 -270 360 180 270 -1 tan(- 110) =
Trig Exact values 6A
WB 12a Exact values 1 60 Some values of Sin, Cos or Tan can be written using fractions, surds, or combinations of both 2 2 60 60 Triangle with sides of length 2 to show this. We can use an Equilateral 2 side in the right angled triangle is 3 (Square root of 22-12) Using Pythagoras, the missing Hyp 30 2 Opp 3 Work out sin, cos and tan of the angles shown 60 1 Opp
WB 12b Exact values 2 60 Opp Hyp 1 2 3 2 2 2 Sin = Sin30 = Sin60 = 60 60 2 Adj Hyp 3 2 1 2 Cos = Cos30 = Cos60 = Hyp 30 2 Opp 3 60 Opp Adj 1 3 3 3 = Tan = Tan30 = Tan60 = 1 Opp 3
WB 12c 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. Hyp 2 Opp 1 hypotenuse will be of length 2 (Square root of 12 + 12) Using Pythagoras Theorem, the 45 1 1 2 2 2 = Sin45 = 1 2 2 2 = Cos45 = 1 1 = 1 Tan45 =
WB 13 Exact values proofs 2 2 =3 4+1 3 1 2 ?) ???230 + ???230 Show that a) ???230 + ???230 = 1 b) sin330 = 1 c) tan60 + 4= 1 = + 2 2 1 1 tan 60= sin 60 cos 60 By symmetry sin 330 = sin( 30) b) Also by symmetry sin ? = sin(?) So sin 30 = sin(30) = 1 2 1 1 3 =4 3 ?) ??? = tan60 + tan60= 3 + 3= 3 + 3 3 1 1 1 4 =4 3 ??? = sin60cos60= = = ??? = ??? 2 1 3 3 3 3 2 4
WB 14 Exact values Given that tan? =3 a) Find sin? and cos? b) solve the simultaneous equations 5?sin? 11 = ? and 3? + 5????? = 1 4 Hyp Iftan? =3 Opp 5 4 3 sin? =3 cos? =4 ? 5 5 4 Opp 5? sin? 11 = ? 3? + 5????? = 1 3? 11 = ? 3? + 4? = 1 3? ? = 11 3? + 4? = 1 3? 11 = 2 3? 8 = 1 ? =3 5? = 10 ? = 2
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
