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# IB SL Trigonometric functions

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### IB SL Trigonometric functions

1. 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. Ene 10­09:53 a.m. 1
2. 2. Lesson 5 Trig. functions 2013 (1).notebook October 29, 2013 Sine & cosine from unit circle by EV.ggb Ene 10­09:53 a.m. 2
3. 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 Ene 10­09:53 a.m. 3
4. 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) Ene 10­09:53 a.m. 4
5. 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. 5
6. 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) Ene 10­09:53 a.m. 6
7. 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 Ene 10­09:53 a.m. 7
8. 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 Ene 10­09:53 a.m. 8
9. 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 Ene 10­09:53 a.m. 9
10. 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 Ene 10­09:53 a.m. 10
11. 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 Ene 10­09:53 a.m. 11
12. 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 Ene 10­09:53 a.m. 12
13. 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| Ene 10­09:53 a.m. 13
14. 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 Ene 10­09:53 a.m. 14
15. 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 Ene 10­09:53 a.m. 15
16. 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 Ene 10­09:53 a.m. 16
17. 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 Ene 10­09:53 a.m. 17
18. 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 Ene 10­09:53 a.m. 18
19. 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| Ene 10­09:53 a.m. 19
20. 20. Lesson 5 Trig. functions 2013 (1).notebook October 29, 2013 Transformations sine.ggb Ene 10­09:53 a.m. 20
21. 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> Ene 10­09:53 a.m. 21
22. 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