trigonometric functions. 1. draw accurate graphs of the functions and 2. draw sketch graphs of and...
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Trigonometric functions
1. Draw accurate graphs of the functions and
2. Draw sketch graphs of andusing their properties
3. Draw accurate graphs of the function 4. Draw a sketch graph of using its properties.5. Investigate the influence of a in the graphs of ,
and 6. Investigate the influence of q on the graphs of ,
and7. Use trigonometric graphs to describe real – life situations.
siny cosy siny cosy
tany tany
siny a cosy a tany a
siny q cosy q tany q
Draw accurate graphs of the functions and
Trigonometric functions are used to describe periodic phenomenon. It is always better to set up a table if you are working with an unknown
function.
0 1 0 - - -1 - 0
siny cosy
siny
0 30 60 90 120 150 180 210 240 270 300 360
3
23
2
1
21
2
1
2
3
23
2
Note:
The sine curve starts at the origin and resembles a wave-like curve.
It takes to complete one cycle and thus the period of the function is
The amplitude of the graph is 1, that is the biggest distance from the position of rest.
The Domain is: x Є R and the range is
The sin function changes, repeats values over a certain interval and is called a periodic function.
360
360
[ 1;1]y
Draw accurate graphs of the functions and
siny cosy
The Graph of the Sine Function
Basic sin curve
y
x
0 30 6090120 150 180 210 240 270300 360
Basic cos curve
y
x
The Graph of the Sine Function
Draw sketch graphs of andusing their properties
siny
cosy
Properties of the sin – function:
•Period =
•The sin – function is a periodic function because there is a repetition of values over a certain interval (check table)•Amplitude = 1
• Range:•The graph starts at ( ;0)• The sin – function is positive in the first and second quadrant.•Use the five points: ,
360
[ 1;1]y
0
( ;0)0 90 ;1 ;(180 ;0); 270 ; 1 and 360 ;1
Draw sketch graphs of andusing their properties
360
[ 1;1]y
Properties of the cos – function: Period =The cos function is a periodic function because there is a repetition
of values over a certain interval (check table)Amplitude = 1Range:The graph starts at The values of cos changes as changes from to The cos – function is positive in the first and fourth quadrant.Use the five points: ,
siny
cosy
0 ;1
0 360
0 ;1 90 ;0 , 180 ; 1 , 270 ;0 and 360 ;1
Sin – function: Using the five points plus the sign in the four quads
y
x
Cos – function: Using the five points plus the sign in the four quads
y
x
Draw accurate graphs of the
function
If:
x- intercepts are at:
Asymptotes at:Amplitude: the tan curve has no amplitude, because it has no maximum or minimum value
tan 45 1 45 ;1
tan135 = -1 135 ;-1
tan 225 1 225 ;1
tan 335 1 315 ;-1
0 , y = 0
x = 90 , y = undefined
x = 180 , y = 0
x = 270 , y = undefined
x = 360 , y = 0
x
tany
0 , x = 180 ; x = 360x
90 and 270 Period of the tan – function = 180º
Range: y Є R
Domain: x Є [ 0º ; 360º ] ; x ≠ ± 90º ;
x ≠ 270º
Test Your Knowledge
1. On the same set of axis sketch the graphs of:
clearly showing the translations that take place.
y=sinx
y=2sinx
y=2sinx-1
Solutions
y
x
The tan - function
x
Draw a sketch graph of using its properties. tany
y
x-
Θ Є [ -90º ; 360º ]
Investigate the influence of a in the graphs of , and
In both the sin and the cos function we see a change in amplitude: amplitude = a unitsperiod = remains range =
siny a cosy a tany a
360[ ; ]y a a
y
x
The transformations in the graph of the cos – function, due to the influence of a
y
x
x
The following graphs clearly show how the graphs of the sin - and cos functions are translated when q is added to the parent graphs
y
x
Use trigonometric graphs to describe real life situations
The heights of tides with respect to time are illustrated in the following sketch. The solid line represents the curve of heights of tides against time.
What is the value of a and q if the equation of the curve is given by
siny a q
2sin 2y The equation is:
y
x
Test your knowledgeQuestion 1 Write down the amplitude and period of the
graph y= - 2 cos2x
Answer
A Amplitude = -2Period = 360°B Amplitude = 2 Period = 180°C Amplitude = -2 Period = 360°D None of the above
Test your KnowledgeQuestion 2
Write down the amplitude and period of the graph y = sin x – 2
Answer
A Amplitude = 1 Period = 360°B Amplitude = 2 Period =360°C Amplitude = - 2 Period = 180°D None of the above
Test your knowledgeQuestion 3
Write down the range and period of the graph y = 2 tanx
Answer
A Range : y ε R Period = 360°B Range : y ε [ - 2 ; 2 ] Period = 360°C Range : y ε R Period = 180°D None of the above
Test your knowledgeQuestion 4
Write down the range of the graph y = 2sinx – 1
Answer
A Range : Range : y ε R B Range : y ε [ - 2 ; 2 ] C Range y ε [ - 1; 3 ]D Range : y ε [ - 3; 1 ]
Test your knowledgeQuestion 5
Write down the amplitude of the graph y = 3tan2x
Answer
A Amplitude = 3B Amplitude = 2C No Amplitude D None of the above
Problems
1. On a certain day the depth, D metres, of water at a fishing port, t hours after midnight, is given by
a. Find the depth of the water at 1.30 pm.b. The depth of the water in the harbour is recorded each hour. What
is the maximum difference in the depths of the water over the 24 hour period?
Solution
pm 1.30at metres 219 isDepth
1d.p. tom219
metres 19322 difference Maximum 2
159512
3)1(59512 of valuemin. )45sin(59512
22159512 of valuemax. Hence, )405sin(59512
1 is minimum ,1)30sin( of valueMaximum b) ))513(30sin(59512,513 When a)
0
D
D
DD
DD
tDt
Problem
Solve 3600,1)sin(240tan 00 xx
Solution
4355or 6184
64- 360or 64180
)64 )08050(sin
)0805.0(sin (r.a.a. 08050)sin(
1610)sin(2
1)sin(28390 so 839040tan
1
1
x
x
x
x
x
x
Trig Graph collection
The following slides give more exemplars of trig graphs. Please study them
0 90 180 360270-90-180-270-360
0 90 180 270-90-180-270-360 360
450o0o 90o 180o 270o 360o-90o-180o-270o-360o-450o
90o
180o
0o 270
o
1
-1
360o
0o
90o
180o
270o
The Trigonometric Ratios for any angle
270-360 90 180
x
y = f(x)
0 360-90-180-270
1
-1
2
-2
sinx
2sinx
3
-3
3sinx
y = ½sinx
90 180
x
0 270-90-180-270
1
-1
2
-2
Sinx
2Sinx
3Sinx
Amplitude 1
Period 360o
Amplitude 2
Period 360o
Amplitude 3
Period 360o
-360
360
-3
3
90 180
x
y = f(x)
0 270 360-90-180-270-360
1
-1
2
-2
3
-3
cosx
½cosx
2cosx
3cosx
xx
x x
y = sinx y = 2sinx
y = 3sinx y = ½ sinx
0
1
2
3
- 1
- 2
- 3
90o 180o 270o 360o-90o-180o-270o-360o 0
1
2
3
- 1
- 2
- 3
90o 180o 270o 360o-90o-180o-270o-360o
0
1
2
3
- 1
- 2
- 3
90o 180o 270o 360o-90o-180o-270o-360o 0
1
2
3
- 1
- 2
- 3
90o 180o 270o 360o-90o-180o-270o-360o
x
x x
y = cosx y = 2cosx
y = 3cosx y = ½ cosx
0
1
2
3
- 1
- 2
- 3
90o 180o 270o 360o-90o-180o-270o-360o 0
1
2
3
- 1
- 2
- 3
90o 180o 270o 360o-90o-180o-270o-360o
0
1
2
3
- 1
- 2
- 3
90o 180o 270o 360o-90o-180o-270o-360o 0
1
2
3
- 1
- 2
- 3
90o 180o 270o 360o-90o-180o-270o-360o x
x
270 360-360
y = f(x)
90 1800-90-180-270
1
-1
2
-2
f(x) = sinx f(x) = sin2x f(x) = sin3x f(x) = sin ½ x
x
270 36090-360 180
y = f(x)
0-90-180-270
1
-1
2
-2
f(x) = cosx f(x) = cos2x f(x) = cos3x f(x) = cos ½ x
xx
x x
y = sinx y = sin2x
y = sin3x y = sin ½ x
0
1
2
3
- 1
- 2
- 3
90o 180o 270o 360o-90o-180o-270o-360o 0
1
2
3
- 1
- 2
- 3
90o 180o 270o 360o-90o-180o-270o-360o
0
1
2
3
- 1
- 2
- 3
90o 180o 270o 360o-90o-180o-270o-360o 0
1
2
3
- 1
- 2
- 3
90o 180o 270o 360o-90o-180o-270o-360o
xx
x x
y =cosx y = cos2x
y = cos3x y = cos ½ x
0
1
2
3
- 1
- 2
- 3
90o 180o 270o 360o-90o-180o-270o-360o 0
1
2
3
- 1
- 2
- 3
90o 180o 270o 360o-90o-180o-270o-360o
0
1
2
3
- 1
- 2
- 3
90o 180o 270o 360o-90o-180o-270o-360o 0
1
2
3
- 1
- 2
- 3
90o 180o 270o 360o-90o-180o-270o-360o
y = tan
0o 90o 180o 270o 360o-90o-180o-270o-360o 450o-450o x
y = sinx
0
1
2
3
- 1
- 2
- 3
90o 180o 270o 360o-90o-180o-270o-360o 0
1
2
3
- 1
- 2
- 3
90o 180o 270o 360o-90o-180o-270o-360o
y = cosx
y = tan
0o 90o 180o 270o 360o-90o-180o-270o-360o 450o-450o x
xx
y = sinx
0
1
2
3
- 1
- 2
- 3
90o 180o 270o 360o-90o-180o-270o-360o 0
1
2
3
- 1
- 2
- 3
90o 180o 270o 360o-90o-180o-270o-360o
y = cosx y = 2sinx y = 2cosx
x
y = 3sinx
0
1
2
3
- 1
- 2
- 3
90o 180o 270o 360o-90o-180o-270o-360o
y = 3cosx
x
y = ½ sinx
0
1
2
3
- 1
- 2
- 3
90o 180o 270o 360o-90o-180o-270o-360o
y = ½ cosx
x x
xx
x x
y = sinx y = -2sinx
y = 3sinx y = ½ sinx
0
1
2
3
- 1
- 2
- 3
90o 180o 270o 360o-90o-180o-270o-360o 0
1
2
3
- 1
- 2
- 3
90o 180o 270o 360o-90o-180o-270o-360o
0
1
2
3
- 1
- 2
- 3
90o 180o 270o 360o-90o-180o-270o-360o 0
1
2
3
- 1
- 2
- 3
90o 180o 270o 360o-90o-180o-270o-360o
y = - sinx y = 2sinx
y = - 3sinx y = - ½ sinx
x
x x
y = cosx y = 2cosx
y = 3cosx y = ½ cosx
0
1
2
3
- 1
- 2
- 3
90o 180o 270o 360o-90o-180o-270o-360o
0
1
2
3
- 1
- 2
- 3
90o 180o 270o 360o-90o-180o-270o-360o
0
1
2
3
- 1
- 2
- 3
90o 180o 270o 360o-90o-180o-270o-360o 0
1
2
3
- 1
- 2
- 3
90o 180o 270o 360o-90o-180o-270o-360o
y = -cosx y = -2cosx
y = -3cosx y = -½ cosx
x
270-360 90 180
x
y = f(x)
0 360-90-180-270
1
-1
2
-2
3
-3
0
1
2
3
- 1
- 2
- 3
90o 180o 270o 360o-90o-180o-270o-360o
0
1
2
3
- 1
- 2
- 3
90o 180o 270o 360o-90o-180o-270o-360o x
x x0
1
2
3
- 1
- 2
- 3
90o 180o 270o 360o-90o-180o-270o-360o
0
1
2
3
- 1
- 2
- 3
90o 180o 270o 360o-90o-180o-270o-360o x