Question

what is the actual value of arc sin (-15/17) + arc sin (-30/36) . use arc sin x + arc sin y formula

Answer 1

Answer on Question #52641 – Math – Trigonometry

Question:

What is the actual value of $\arcsin(-15/17) + \arcsin(-30/36)$? Use $\arcsin x + \arcsin y$ formula.

Solution:

$$\begin{aligned} \arcsin x + \arcsin y &= \\ & \arcsin \left(x \sqrt{1 - y^2} + y \sqrt{1 - x^2}\right), \text{ if } xy \leq 0 \text{ or } x^2 + y^2 \leq 1, \\ & = \arcsin \left(x \sqrt{1 - y^2} + y \sqrt{1 - x^2}\right), \text{ if } x > 0, y > 0 \text{ and } x^2 + y^2 > 1, \\ & - \pi - \arcsin \left(x \sqrt{1 - y^2} + y \sqrt{1 - x^2}\right), \text{ if } x < 0, y < 0 \text{ and } x^2 + y^2 > 1. \end{aligned}$$

Because $x = -\frac{15}{17} < 0$, $y = -\frac{30}{36} < 0$ and

$$x^2 + y^2 = \left(-\frac{15}{17}\right)^2 + \left(-\frac{30}{36}\right)^2 = \frac{15325}{10404} \approx 1.473 > 1, \text{ then we apply}$$ $$\begin{aligned} \arcsin x + \arcsin y &= -\pi - \arcsin \left(x \sqrt{1 - y^2} + y \sqrt{1 - x^2}\right) \\ &= -\pi - \arcsin \left(\left(-\frac{15}{17}\right) \sqrt{1 - \left(-\frac{30}{36}\right)^2} + \left(-\frac{30}{36}\right) \sqrt{1 - \left(-\frac{15}{17}\right)^2}\right) \\ &= -\pi - \arcsin \left(-\frac{240 + 90\sqrt{11}}{17 * 36}\right) \\ \end{aligned}$$

approximately $-241.63{}^\circ$.

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