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:

$\cos(x+90)^{\circ}=?$
$\sin(x-90)^{\circ}=?$
$\sin(x+90)^{\circ}=?$

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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). Remember to use the degree symbol $^{\circ}$ . A pc shortcut is ALT O; a mac short cut is SHIFT OPTION 8.

Graph 1:

Graph 2:

Graph 3:

Graph 4:

Graph 5:

Graph 6:

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