Trigonometric Graphs and Patterns
C
2
:
T
r
i
g
o
n
o
m
e
t
r
y
•
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
S
t
a
r
t
e
r
:
c
a
l
c
u
l
a
t
e
What patterns can
you see?
WB 10 Trig graphs
You should know
these intimately!
1
-1
90º
180º
270º
360º
y
θ
y
90º
180º
270º
360º
-90º
-180º
-360º
1
0
-1
-270º
θ
y = sin
θ
y = cos
θ
y = tan
θ
90º
180º
270º
360º
-90º
-180º
-360º
1
0
-1
-270º
θ
0
Describe the
features of each
to your neighbour
-
Similarities
-
Intersections
with the axes
-
Period of each
graph
-
Asymptotes on
the tan graph
https://www.geogebra.org/m/S2gMrkbD
WB 11a Symmetry of sin graph
You need to be able to
recognise the graphs of sin
θ
,
cos
θ
and tan
θ
You need to be able to work out
larger values of sin, cos and tan
as acute angles (0º - 90º)
Write sin 130º as sine of an
acute angle
(sometimes asked as a
‘trigonometric ratio’)
Sin 130º = Sin 50º
y = sin
θ
1
90º
180º
270º
360º
y
θ
0
-1
130
50
-40
-40
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 11b Symmetry of cos graph
+30
+30
You need to be able to
recognise the graphs of sin
θ
,
cos
θ
and tan
θ
You need to be able to work out
larger values of sin, cos and tan
as acute angles (0º - 90º)
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)
y = cos
θ
-270º
y
1
0
-1
θ
90º
180º
270º
-180º
-120
-60
60
-90º
+60
+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
θ
1
0
-1
θ
WB 11d More Symmetry of trig graphs
F
o
r
a
n
y
a
n
g
l
e
ϴ
,
s
i
n
(
-
ϴ
)
=
-
s
i
n
ϴ
c
o
s
(
-
ϴ
)
=
c
o
s
ϴ
t
a
n
(
-
ϴ
)
=
-
t
a
n
ϴ
C
o
m
p
l
e
t
e
:
s
i
n
(
-
2
5
)
=
c
o
s
(
-
2
4
0
)
=
t
a
n
(
-
1
1
0
)
=
6A
WB 12a
Exact values 1
2
1
√3
60˚
30˚
Opp
Hyp
Opp
Some values of Sin, Cos or Tan can be
written using fractions, surds, or
combinations of both…
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 2
2
-1
2
)
Work out sin, cos and tan of the
angles shown
WB 12b Exact values 2
60˚
60˚
60˚
2
2
2
2
1
√3
60˚
30˚
Opp
Hyp
Opp
Opp
Hyp
Sin
θ
=
1
2
Sin30
=
√3
2
Sin60
=
Adj
Hyp
Cos
θ
=
√3
2
Cos30 =
1
2
Cos60 =
Opp
Adj
Tan
θ
=
1
√3
Tan30
=
√3
Tan60 =
√3
3
=
WB 12c Exact values 3
Opp
Hyp
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 1
2
+ 1
2
)
1
√2
Sin45 =
√2
2
=
1
√2
Cos45 =
√2
2
=
1
1
Tan45 =
1
=
WB 13 Exact values proofs
WB 14 Exact values
self-assess
One thing learned is –
One thing to improve is –
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
END
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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Presentation Transcript
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
