• Join over 1.2 million students every month
  • Accelerate your learning by 29%
  • Unlimited access from just £6.99 per month

Math Investigation - Sine Law

Extracts from this document...

Introduction

Part 1

Look at the graphs of y = sin x

Compare the graphs of:

y = sin x
image00.jpg

image01.jpg

y = 2sin x

y = 1/3sin ximage02.jpg

image03.jpg

y = 5sin x

The difference between all these graphs is a variable known as A, or amplitude of wave.

When A > 1, the graph stretches vertically.

When 0 < A < 1, the graph compresses vertically.

Also, A is the number that manipulates how far the graph compresses or stretches to. For example, in the graph of y = 2sin x, the graph stretches out to +2 and down to -2. The characteristics of the waveform are altered because of this. The range of the graph is increased or decreased in conclusion.

The domain and range of the graphs are:

y = sin x

D: {x| xεR}

R: {y| -1 < y < 1, yεR}

y = 2sin x

D: {x| xεR}

R: {y| -2 < y < 2, yεR}

y = 1/3sin x

D: {x| xεR}

R: {y| -1/3 < y < 1/3, yεR}

y = 5sin x

...read more.

Middle

y = sin x?

The C variable in y = sin (x + C) is a variable that creates a horizontal translation.

In example, if C is substituted with 90, the equation becomes y = sin (x + 90):

image05.jpg

y = sin x

y = sin (x+90)

It is evident that the entire graph shifted to the left by 90 unit. This can conclude that the amount that the graph moves by is by C, except it moves the direction opposite to the symbol in front of it. In this case, C, 90, caused the graph to move to the left by 90. This said, if C was -90, the graph would move to the right by 90 units. The range for y = sin (x + C) is not affected by the translation.

The domain and range of the graphs are:

y = sin x

D: {x| xεR}

R: {y| -1 < y < 1, yεR}

y = sin (x+90)

D: {x| xεR}

R: {y| -2 < y < 2, yεR}


Part 3

...read more.

Conclusion

 + 90o)

I predicted this function as a vertical stretch since A > 1. The C in the graph translates the line 90 units to the left. There is no D in this function.

D: {x| xεR}

R: {y| -3 < y < 3, yεR}

y = ½sin x + 3

I predicted this function as a vertical compression because A < 1. There is no C present in this function. The D in the function moves the line up 3 units on the y-axis.

D: {x| xεR}

R: {y| -2½ < y < 3½, yεR}

Part 5

The graph of y = cos xand the graph of y = sin xis almost the exact same thing. The only difference is that the same graph would be translated 90o to the left if it involved the cos function.

Essentially, cos (x) = sin (x +90o).

Since this is true, it is conclusive that sine is a very versatile function with many altering variables.

image08.jpg

y = cos x

y = sin (x + 90o)

*Although it is not completely evident, the two lines are overlapping, as I had predicted.

...read more.

This student written piece of work is one of many that can be found in our International Baccalaureate Maths section.

Found what you're looking for?

  • Start learning 29% faster today
  • 150,000+ documents available
  • Just £6.99 a month

Not the one? Search for your essay title...
  • Join over 1.2 million students every month
  • Accelerate your learning by 29%
  • Unlimited access from just £6.99 per month

See related essaysSee related essays

Related International Baccalaureate Maths essays

  1. Extended Essay- Math

    �_� tIME � 2-�� IDATx��lg�Ç.H��J� $�Ur�v˪�Zicʪ���u���Õ��:�I�p"'�u�n7�S�d"m"��%]�$T'J...���Ӷ8�c�"�بEvUP-"8ZV�hs�d�(tm)_-{Æ���(���x�_�y��}ç��E�\ܧ�P-@.@�KP��E@.@�KP�QSï¿½ï¿½Ô g:"�� ����^���"4?c��dK�^1/2�" �@�KP��E@.@�KP��E@.@U�T(�#�[�ն֦�!����l - ��b�C��-��6T5�A��-��X�� Ú\�2 v5����� P�@.A�&�� �"(c)�"�j�� P�@.��I�L�����%0<LÇ¥-@ -�� ��%0: r �N�"c3�����5�@.��Igҵ��z[�Q��~�{���f�9�%x(r)-�H�z�n�,"` *-"OF�Q�L�E����d2I~%��-f96B.���0��<;<���mK�Dc��]͵��'P"��'r9��X)��f����1/4 I�L�h�&P=f"���Cz�r�X�jT�",R�����^� � O...{��R�:~Ñ¡"(tm)jI^On��V�6��w3v{�{i�yw$-���TZ~�r�+�-�s�'ߣ����^�H��̴<�B���ȳ�Ûe(r)D�4=ִ�d��b���N/(tm)!�^�*j"$�T�m���f+(c)}����|�^W-���-Ø{�^tlvpU��1/4 1�\r\��E�^1/2��c"�XGUBH�"h(tm)j ���6��'���㥵K3�P���k�uuu�K|u'`|�Ҵ}ò-�3/4�-;=���V{��(tm)8*8...���t���"�n*�ض1/2mh464���(���j�f2(tm)D"!c��/g%�yW3×R#@�$���w��<�(c)ʪ�6��C�D�q��� ^�=�[�U...sg�j���*Z�I.M�6 ���l��kn9V� ]�?m����-J����A���\"F���\}vv�mO[z.�x�l�(c)( ��`1/4aK�y��S�7g) �=��JTH1/4N(�W�ׯ_?(tm)�6�����w\���C�hO�6���ݢ-hx�a��"T(r)�k��c1/2c�ƺ~(c)v�'��gYc!"�_FmvWrt���n��k��8Px�(r)�]NPUSc��!G�B(�i�2�;��rT� &z-��(r��(r)�e��'Õ�FA^B�n�<��TXs����MPAX,(r){T� (��O "�e����H�+]��� J�B͹�T���lx���]��r%zEt�{;|'c Q���8��"~��......����o�3/4�3/4'wG��-�/�V�_�N�$-�5�8�R��>�.Τ��An��k�...�BγQ ��Q<-�m��s�3(tm)A�so���... �n�5LM�SX� �}xn�' ;"�3�9)Y��� jk�� �S�� j9�Y�����=�="!

  2. Investigating Sin Functions

    Again, we have to use Absolute Value. This leads to my conjecture. Conjecture - Part 2 When |b|>1, the period of the function will be less than 2?, and the graph will be horizontally shrunk (making the wave shorter) by a factor of 1/|b|. And when |b| < 1, the period of the function will be greater than 2??

  1. derivitaive of sine functions

    Therefore a limits proof will be made to determine the derivative of the function. We will first begin with the general formula used to determine the limits or the derivative of a function.

  2. Function Transformation Investigation

    Fortunately, trigonometric functions do this for us. These two graphs not only confirm that multiplying the output or input stretches the graph but it also helps us understand how to calculate the stretch and also the direction. The first graph (cosine graph)

  1. Investigating the Graphs of Sine Function.

    This conjecture can be expressed in terms of transformations because it can be noticed by looking at all the graphs shown above, that all of them are transformations of the base graph y = sin x. For example, the transformation of the graph y = sin x into curve y

  2. Derivative of Sine Functions

    is: f (x) = cosx Question 2 Investigate the derivatives of functions of the form f (x)=asinx in similar way a) Consider several different values of a. Consider three values as a=-1,a=2 and a=-2 , 1. When 1 a=-1 f(x) =-sinx. 2 a=-2 f(x)

  1. Verify Newtons second law.

    Time taken for wooden block to pass GATE 1 ( � 0.000002 s) ( T1 - t1 ) Time take for wooden block to pass GATE 2 ( � 0.000002 s)

  2. Music and Maths Investigation. Sine waves and harmony on the piano.

    They are given the symbols of â¯, which means sharp and â­, which means flat. Here is an image with the keyboard and its representational letters: Image Source: http://www.piano-keyboard-guide.com/piano-keyboard-layout.html Each semitone, or note, will produce a sound that has its own frequency.

  • Over 160,000 pieces
    of student written work
  • Annotated by
    experienced teachers
  • Ideas and feedback to
    improve your own work