Let the dual scalar ẑ = (φ, d) define a dual angle, then the infinite series definitions of sine and cosine yield the relations sin ⁡ ⁡ z ^ ^ = ( sin ⁡ ⁡ φ φ , d cos ⁡ ⁡ φ φ ) , cos ⁡ ⁡ z ^ ^ = ( cos ⁡ ⁡ φ φ , − − d sin ⁡ ⁡ φ φ ) , which are also dual scalars. In general, the function of a dual variable is defined to be f (ẑ) = (f (φ), df ′(φ)), where df ′(φ) is the derivative of f (φ).