2.8.5 $\arcsin x$ 与 $\arcsin y$ 的和与差

\arcsin x + \arcsin y

\begin{matrix} (2.155a) & = \arcsin (x \sqrt{1 - y^{2}} + y \sqrt{1 - x^{2}}) (x y \leq 0 或 x^{2} + y^{2} \leq 1) \end{matrix}

= π - \arcsin (x \sqrt{1 - y^{2}} + y \sqrt{1 - x^{2}}) (x > 0, y > 0, x^{2} + y^{2} > 1)

(2.155b)

= - π - \arcsin (x \sqrt{1 - y^{2}} + y \sqrt{1 - x^{2}}) (x < 0, y < 0, x^{2} + y^{2} > 1) .

(2.155c)

\arcsin x - \arcsin y

\begin{matrix} (2.156a) & = \arcsin (x \sqrt{1 - y^{2}} - y \sqrt{1 - x^{2}}) (x y \geq 0 或 x^{2} + y^{2} \leq 1) \end{matrix}

\begin{matrix} (2.156b) & = π - \arcsin (x \sqrt{1 - y^{2}} - y \sqrt{1 - x^{2}}) (x > 0, y < 0, x^{2} + y^{2} > 1 \end{matrix}

= - π - \arcsin (x \sqrt{1 - y^{2}} - y \sqrt{1 - x^{2}}) (x < 0, y > 0, x^{2} + y^{2} > 1) .

(2.156c)

2.8.5 arcsin⁡x 与 arcsin⁡y 的和与差 ​