Question

Evaluate: sin⁡[cos^(-1)⁡((-2)/7) ]

a. √53/7 b. (-√53)/7 c. (3√5)/7 d. (-3√5)/7 e. None of these

Answer 1

Answer on Question #73819 – Math – Trigonometry

Question

Evaluate $\sin \left[\cos^{-1}\left(\frac{-2}{7}\right)\right]$

$$a.\frac{\sqrt{53}}{7} b.\frac{-\sqrt{53}}{7} c.\frac{3\sqrt{5}}{7} d.\frac{-3\sqrt{5}}{7} e.\text{None of these}$$

Solution

Let $\varphi = \cos^{-1}\left(\frac{-2}{7}\right)$ then $\cos(\varphi) = \frac{-2}{7}$, hence $\frac{\pi}{2} \leq \varphi \leq \pi$.

Find $\sin(\varphi): \sin(\varphi) = \pm \sqrt{1 - \cos^2(\varphi)} = \pm \frac{3\sqrt{5}}{7}$

Because of $\pi/2 \leq \cos^{-1}\left(\frac{-2}{7}\right) \leq \pi$, then $\sin \left[\cos^{-1}\left(\frac{-2}{7}\right)\right] = \frac{3\sqrt{5}}{7}$

Answer: $c.\frac{3\sqrt{5}}{7}$

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