1. Lesson 5 Trig. functions 2013 (1).notebook
October 29, 2013
Trigonometric Functions Revisited
• Recognise the graphs of sine, cosine and
tangent functions, their periods and amplitudes.
• Revise the geometrical transformations of
trigonometric functions.
• Draw sketches of the trigonometric functions.
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2. Lesson 5 Trig. functions 2013 (1).notebook
October 29, 2013
Sine & cosine from unit circle by EV.ggb
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3. Lesson 5 Trig. functions 2013 (1).notebook
October 29, 2013
Sine function
x
0
π/6
sin x
π/3
π/2
2π/3
5π/6
π
3π/2
2π
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4. Lesson 5 Trig. functions 2013 (1).notebook
October 29, 2013
Sine function
period
amplitude
maximum value = 1
minimum value = -1
amplitude= 1
range= [-1,1]
period= 2π ( or 360o)
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5. Lesson 5 Trig. functions 2013 (1).notebook
October 29, 2013
Cosine function
x
0
π/6
cos x
π/3
π/2
2π/3
5π/6
π
3π/2
2π
Sine & cosine from unit circle by EV.ggb
Ene 10­09:53 a.m.
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6. Lesson 5 Trig. functions 2013 (1).notebook
October 29, 2013
Cosine function
period
amplitude
maximum value = 1
minimum value = -1
amplitude= 1
range= [-1,1]
period= 2π ( or 360o)
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7. Lesson 5 Trig. functions 2013 (1).notebook
October 29, 2013
Tangent function
Tan from unit circle by EV.ggb
FromCircleToSineAndCosineCurvesWithAngleInDegrees.nbp
http://www.ies.co.jp/math/products/trig/applets/graphTanX/graphTanX.html
http://www.analyzemath.com/unitcircle/unitcircle.html
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8. Lesson 5 Trig. functions 2013 (1).notebook
October 29, 2013
Tangent function
period
vertical asymptotes: x = π/2 , x = 3π/2, x = 5π/2,....
period : π
no amplitude
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9. Lesson 5 Trig. functions 2013 (1).notebook
October 29, 2013
Transformations of the trigonometric functions
in [­2π, 2π]
y = sin x + 2
y = sin (x + 2)
y = sin (2x)
y =2 sin x
y = sin (­x)
y = ­ sin x
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10. Lesson 5 Trig. functions 2013 (1).notebook
October 29, 2013
Vertical translations
y = sin x +2
y = sin x
y = sin x ­1
y = sin x + c
y = cos x + c
represent vertical translations of vector
y = tan x + c
period and amplitude don't change
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11. Lesson 5 Trig. functions 2013 (1).notebook
October 29, 2013
Horizontal translations
y = sin x
y = cos x
period and amplitude don't change
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12. Lesson 5 Trig. functions 2013 (1).notebook
October 29, 2013
Horizontal translations
y = sin ( x ±a )
y = cos ( x ±a )
y = tan ( x ±a )
represent horizontal translations
of the curves y = sinx, y = cos x and
y = tan x
y = sin ( x - a )
to the right
y = sin ( x + a )
to the left
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13. Lesson 5 Trig. functions 2013 (1).notebook
October 29, 2013
Vertical dilation (stretch)
y =2 cos x
y = cos x
y = sin x
y = a sin x
y = a cos x are dilations of the curves y = cos x
y = a tan x
y = tan x
parallel to
the y-axis
the amplitude changes to |a|
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14. Lesson 5 Trig. functions 2013 (1).notebook
October 29, 2013
Horizontal dilation (stretch)
y = sin x
y = sin (2x)
period : π
horizontal stretch , scale factor ½
period changes to half the original
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15. Lesson 5 Trig. functions 2013 (1).notebook
October 29, 2013
period= 6 π
y = sin x
period= 2 π
horizontal stretch , scale factor 3
period changes to three times the original
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16. Lesson 5 Trig. functions 2013 (1).notebook
y = sin (bx)
are dilations of the curves
y = cos(b x)
October 29, 2013
y = sin x
y = cos x
parallel to
the x-axis
the period changes to
y = tan( bx)
is a dilation of the curve
y = tan x
parallel to
the x-axis
the period changes to
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17. Lesson 5 Trig. functions 2013 (1).notebook
October 29, 2013
Reflections
reflection in the x-axis
y = sin x
y = ­ sin x
y = tan(­x)
y = tan x
reflection in the y-axis
period and amplitude don't change
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18. Lesson 5 Trig. functions 2013 (1).notebook
October 29, 2013
Combined transformations
y= a sin [b(x­ c)]+ d
vertical translation
horizontal translation
horizontal dilation: period =
amplitude= |a| (vertical dilation)
Transformations sine.ggb
the same applies for cosine function
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19. Lesson 5 Trig. functions 2013 (1).notebook
October 29, 2013
Combined transformations
y= a tan [b(x ­ c)]+ d
vertical translation
horizontal translation
horizontal dilation: period =
amplitude= |a|
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20. Lesson 5 Trig. functions 2013 (1).notebook
October 29, 2013
Transformations sine.ggb
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21. Lesson 5 Trig. functions 2013 (1).notebook
October 29, 2013
Book Ex 8D 8 E
Other related websites you can visit
http://members.shaw.ca/ron.blond/sc.APPLET/index.html
http://www.ies.co.jp/math/products/trig/applets/graphSinX/graphSinX.html
http://magnus­karlsson.nu/mathplets/trig/trigfunc.html
http://www.mathsnet.net/asa2/modules/p13unitcircle3.html
<http://207.207.4.198/pub/flash/15/graphing.swf>
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22. Adjuntos
functions transformations.nbp
Sine & cosine from unit circle by EV.ggb
Transformations sine.ggb
FromCircleToSineAndCosineCurvesWithAngleInDegrees.nbp
Tan from unit circle by EV.ggb