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常用曲线极坐标方程及其图形⚓︎

积化和差与和差化积恒等式⚓︎

积化和差 和差化积
sin⁡αcos⁡β=sin⁡(α+β)+sin⁡(α−β)2\sin\alpha\cos\beta = \dfrac{\sin(\alpha + \beta) + \sin(\alpha - \beta)}{2} sin⁡α+sin⁡β=2sin⁡α+β2cos⁡α−β2\sin\alpha + \sin\beta = 2\sin\dfrac{\alpha + \beta}{2}\cos\dfrac{\alpha - \beta}{2}
cos⁡αsin⁡β=sin⁡(α+β)−sin⁡(α−β)2\cos\alpha\sin\beta = \dfrac{\sin(\alpha + \beta) - \sin(\alpha - \beta)}{2} sin⁡α−sin⁡β=2cos⁡α+β2sin⁡α−β2\sin\alpha - \sin\beta = 2\cos\dfrac{\alpha + \beta}{2}\sin\dfrac{\alpha - \beta}{2}
cos⁡αcos⁡β=cos⁡(α+β)+cos⁡(α−β)2\cos\alpha\cos\beta = \dfrac{\cos(\alpha + \beta) + \cos(\alpha - \beta)}{2} cos⁡α+cos⁡β=2cos⁡α+β2cos⁡α−β2\cos\alpha + \cos\beta = 2\cos\dfrac{\alpha + \beta}{2}\cos\dfrac{\alpha - \beta}{2}
sin⁡αsin⁡β=−cos⁡(α+β)−cos⁡(α−β)2\sin\alpha\sin\beta = -\dfrac{\cos(\alpha + \beta) - \cos(\alpha - \beta)}{2} cos⁡α−cos⁡β=−2sin⁡α+β2sin⁡α−β2\cos\alpha - \cos\beta = -2\sin\dfrac{\alpha + \beta}{2}\sin\dfrac{\alpha - \beta}{2}

降幂公式⚓︎

正弦 余弦
sin⁡2θ=1−cos⁡2θ2\sin^2\theta = \dfrac{1 - \cos2\theta}{2} cos⁡2θ=1+cos⁡2θ2\cos^2\theta = \dfrac{1 + \cos2\theta}{2}
sin⁡3θ=3sin⁡θ−sin⁡3θ4\sin^3\theta = \dfrac{3\sin\theta - \sin3\theta}{4} cos⁡3θ=3cos⁡θ+cos⁡3θ4\cos^3\theta = \dfrac{3\cos\theta + \cos3\theta}{4}

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