List of Trigonometric Identities - Certain Linear Fractional Transformations

Certain Linear Fractional Transformations

If ƒ(x) is given by the linear fractional transformation

and similarly

then

f(g(x)) = g(f(x)) = \frac{(\cos(\alpha+\beta))x - \sin(\alpha+\beta)}{(\sin(\alpha+\beta))x + \cos(\alpha+\beta)}.

More tersely stated, if for all α we let ƒα be what we called ƒ above, then

If x is the slope of a line, then ƒ(x) is the slope of its rotation through an angle of −α.

Read more about this topic:  List Of Trigonometric Identities

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