Quadruple Angle Identities
Quadruple angle identities give relationship between trigonometric functions of an angle and the one that is four multiples of the angle.
There are 6 triple angle identities corresponding to 6 elementary trigonometric functions. However, the most frequent used ones are for sines, cosine and tangent functions. Quadruple implies repeating use of double angle identities when it comes to derive the Quadruple Angle Identities.
\sin2 \alpha = 2\sin \alpha \cos \alpha
\cos2 \alpha = 2\cos^2\alpha -1
\cos2 \alpha = 1-2 \sin^2 \alpha
\tan 2 \alpha = \dfrac{2\tan \alpha }{1-\tan^2\alpha }
Quadruple Angle Identity for Sine function
\begin{aligned} \sin4\theta& = \sin 2\cdot 2\theta \\ &= 2\sin 2\theta \cdot \cos 2\theta \\ &=4\sin \theta \cos\theta (1-2\sin^2\theta ) \\ & =4\sin\theta\cos\theta -8\sin^3\theta \cos\theta \end{aligned}
Noticed \sin4\theta is not represented in the form of the same function.
\sin4\theta = 4\sin\theta\cos\theta -8\sin^3\theta \cos\theta
(1)
Quadruple Angle Identity for Cosine function
\begin{aligned} \cos 4\theta & = \cos2\cdot 2\theta \\ &=2 \cos^2 2\theta -1 \\ &=2(2\cos^2 \theta -1 )^2 \\& =8\cos^4\theta -8\cos^2\theta +1 \end{aligned}
\cos 4\theta could be converted to a quartic expression in terms of \cos\theta .
\cos 4\theta = 8\cos^4\theta -8\cos^2\theta +1
(2)
Quadruple Angle Identity for Tangent function
\begin{aligned} \tan 4\theta & = \tan2\cdot 2\theta \\ &=\dfrac{2\tan2\theta }{1-\tan^2 2\theta } \\ &=\dfrac{2\cdot \dfrac{2\tan\theta }{1-\tan^2\theta } }{1 - (\dfrac{2\tan\theta }{1-\tan^2\theta } )^2} \\& = \dfrac{4\tan\theta -4\tan^3\theta }{1-6\tan^2\theta +\tan^4\theta } \end{aligned}
\tan 4\theta could be transformed to a fractional expression in terms of \tan\theta .
Steven Zheng
posted 5 months ago