Lets consider the basic trig functions: sinx and cosx

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Lets consider the basic trig functions sin(x)
and cos(x)
0 x 2p
Important points to remember Initial values of
these functions, when x 0, are sin(x) 0 and
cos(x) 1. Range of values of the functions
sin(x) and cos(x) changes between -1 and 1. The
period of the functions sin(x) and cos(x) is 2?
in radians (or 3600 in degrees).
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Frequency change

• y sin(x) Period T 2?
• y sin(2x) Period T 2? / 2

• y sin(x) Period T 2?
• y sin(3x) Period T 2? / 3

sin(x)
In general y sin(Bx) Period
T 2? / B or B 2 ? / T
Remember The number that multiply by x inside
the trig-function is responsible for frequency
change.
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Change in amplitude

• y sin(x) Period T 2?
  Range -1 y 1
• y 2sin(x) Period T 2?
  Range -2 y 2

1. y sin(x) Period T 2?
Range -1 y
1 2. y 3sin(x) Period T 2?
Range -3 y
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In general y Asin(x) Period T
2?
Range - A y A
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Vertical shift (y-shift)

1. y sin(x) Period T 2?
Range -1 y 1 2. y
sin(x) 1 Period T 2?
Range yshift - 1 y yshift 1
1. y sin(x) Period T 2?
Range -1 y 1 2. y
sin(x) - 2 Period T 2?
Range yshift - 1 y yshift 1
sin(x)
In general y sin(x) D or y sin(x)
y-shift
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Horizontal shift (x-shift)
1. y sin(x) 2. Y sin(x ?/4)
1. y sin(x) 2. Y sin(x - ?/4)
In general y sin(x - C) or y sin(x
xshift)
Remember The number that is subtracted from x
inside the trig-function is responsible for the
shift across x axis.
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General formula
General equation can be written as y
Asin(B(x - C)) D
Or y A sin(B(x - xshift)) yshift
Useful formulae Amplitude Frequency
change
or y-shift X-shift figure out from
the given initial conditions (x 0)
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Hints for generating the trig equation from the
given conditions 1. Choose sin or cos function
according to the initial condition of the given
task. E.g. If Maximum value of a function
corresponds to the initial situation (x0), then
choose cos function. If Minimum value of a
function corresponds to the initial situation
(x0), then choose -cos function. If
zero corresponds to the initial situation (x
0), then choose sin function. If Min, Max,
or zero value occurs before or after initial
situation, then x-shift should be applied. 2.
If the Minimum and Maximum value of a function is
given, figure out the Amplitude and y-shift
using the formulae
y A sin(B(x - xshift)) yshift
or
y Asin(B(x - C)) D
3. If a period is given, then figure out the
frequency
4. Write down an expression of the function and
plot it on the graphics calculator.
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Hints for plotting trig functions on the graphics
calculator follow 3 simple steps
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Merit questions
NCEA 2008 External exam - Y12 Trig 2.9 (90292)
SOLUTION
Step 1 set up correct units on your graphics
calculator - MENU RUN ShiftMenu (Set Up)
scroll down to highlight Angle F2 (Rad) - Exit
Step 2 Set up correct settings for x and y
axes Shift F3 (V-Window) F2 (Trig) then
change setting according to the task conditions
Xmin 0 Xmax 7 (key words in the
task are a week and t is the time in days)
scale 1 (choose sensibly) Ymin
2.5 -1 (y-shift A ) Ymax 2.5 1
(y-shift A) Scale 0.5
(choose scale sensibly)
Step 3 Plot the graph MENU GRAPH y1
cos(0.8x) 2.5
y2
2.65 then find the two
intercepts F5(GSolve) F5 (ISCT)


(use REPLEY button to find the next
root)
Step 4 Find the interval when the weight is
less than 2.65 kg. This is the difference
between the two solutions
Answer 6.08 1.78 4.3 days
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Merit questions
SOLUTION
Step 2 Settings for x and y axes Shift F3
(V-Window) F2 (Trig) Xmin 0 Xmax 24
(key words in the task are 24-hour
period and t is the number of hours) scale
1 (choose sensibly) Ymin 15 -1
(y-shift A ) Ymax 15 1
(y-shift A) Scale 1 (choose scale
sensibly)
Step 3 Plot the graph MENU GRAPH y1
15 sin(?x/12)
y2
14.6 then find the two intercepts
F5(GSolve) F5 (ISCT)

(use
REPLEY button to find the next root)
Step 1 set up correct units on your graphics
calculator - MENU RUN ShiftMenu (Set Up)
scroll down to highlight Angle F2 (Rad) - Exit
Step 4 Find for how long the temperature is
above 14.6. You can calculate the interval when
the temperature is below 14.6 and then subtract
it from 24 hours.
Answer 22.43 13.57 8.86
24 8.86 15.14 hours.
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Excellence question
SOLUTION
Step 3 plot the graph y1 1200
400sin(?x/6)
y2 1500
and find the two intercepts. Answer is the
difference between the two solutions 10.38
7.62 2.76 months.
Step 2 Write down the equation and figure out
the settings for x and y to plot the function on
the graphics calculator
y 1200 400sin(?x/6)
Settings Xmin 0
Xmax 12 (key words Period
of the model is 12 months)
scale 1
Ymin 1200 400 (y-shift
amplitude) Ymax
1200 400 (y-shift amplitude)
scale 100
(choose sensibly)
Step 1 Find the coefficients A, B and C.
Use the initial value to find A when t
0, R A 1200. The
difference between the maximum and the initial
value is an amplitude B 1600 1200 400
Use period (T 12) to find C
C 2?/12 or C ?/6
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Merit question
Y13 Trigonometry - KTI resources "Sunshine"
SOLUTION
(a) Choose a trig-function key words the
lowest temperature was at 3am give us a clue
that it is better to use negative cos
function. Therefore, the equation
can be written as y D Acos(B(x C))
where C is the x-shift and equals
3, because the lowest temperature was at 3am.
B is frequency change
B 2?/24 or B ? /12
(because the period is 24)
A is an amplitude A (max
min)/2 (29 7) / 2 11
D is the y-shift D (max
min) / 2 (29 7) / 2 18 Now we can write
down the equation y 18 11cos(?(x 3) /
12)
(c) At midnight x 0. Then y 18
11cos(-3 ? / 12) 10.22
(d) At mid day x 12. Then y 18
11cos(?(12-3) / 12) 25.78
(e) Settings for the graphics calculator
Xmin 0 Xmax 24
(Period is 24 hours) Scale 6
(you can choose different scale) Ymin
18 11 (y-shift amplitude)
Ymax 18 11 (y-shift amplitude)
Scale 1 Plot the graph y1 18
11cos(?(x 3) / 12)
y2 20
and find the intercept.
(b) The mathematical model may not describe the
real situation perfectly, because the gradient of
increasing and decreasing the temperature may be
different. Therefore the trig-function will be
only approximation .