Sine and Cosine Transformations

When identifying a sinusoidal function, you may use the basic shape of either the sine curve or the cosine curve – one is just a horizontal transformation of the other!

The curve in the figure above can take the equation y = 3\sin(x^{\circ}) or y = 3\cos(x-90)^{\circ}.

Use the applet to view the four basic shapes.

Exercise 1:

It has been noted above that \cos(x-90)^{\circ}=\sin(x^{\circ}). In other words, if you slide the cosine function 90 units to the right, your curve can be expressed as the sine function.

In a similar way, complete:

  1. \cos(x+90)^{\circ}=? (Move the cosine curve 90 units left, you may do this on the applet just reset it when done to avoid confusion.)
  2. \sin(x-90)^{\circ}=?
  3. \sin(x+90)^{\circ}=?

When a curve takes one of these four basic shapes shown in the applet, we can avoid using a horizontal transformation.

Exercise 2:

Reproduce the following graphs on geogebra; using only functions of the form y=a\sin(bx^{\circ})+d or y=a\cos(bx^{\circ})+d (that is, no horizontal translation).

Graph 1:

Graph 2:

Graph 3:

Graph 4:

Graph 5:

Graph 6:


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