Q92 / 180 · COMEDK 2024 (Morning Shift) · Mathematics
Evaluate : $\cos ^{-1}\left[\cos \left(-680^{\circ}\right)\right]+\sin ^{-1}\left[\sin \left(-600^{\circ}\right)\right]-\cos ^{-1}\left(\sin 270^{\circ}\right)$
- A$\pi$
- B$\dfrac{14 \pi}{9}$
- Correct answerC$-\dfrac{4 \pi}{9}$
- D$-\dfrac{5 \pi}{9}$
Solution
The expression is $E = \cos^{-1}(\cos(-680^{\circ})) + \sin^{-1}(\sin(-600^{\circ})) - \cos^{-1}(\sin 270^{\circ})$.
First, simplify the angles:
$\cos(-680^{\circ}) = \cos(680^{\circ}) = \cos(720^{\circ} - 40^{\circ}) = \cos(-40^{\circ}) = \cos(40^{\circ})$.
Since $40^{\circ} \in [0, 180^{\circ}]$, $\cos^{-1}(\cos 40^{\circ}) = 40^{\circ}$.
Next, $\sin(-600^{\circ}) = -\sin(600^{\circ}) = -\sin(720^{\circ} - 120^{\circ}) = -(-\sin 120^{\circ}) = \sin 120^{\circ} = \sin(180^{\circ} - 60^{\circ}) = \sin 60^{\circ}$.
Since $60^{\circ} \in [-\pi/2, \pi/2]$, $\sin^{-1}(\sin 60^{\circ}) = 60^{\circ}$.
Finally, $\sin 270^{\circ} = -1$.
$\cos^{-1}(-1) = 180^{\circ}$.
Substituting these values into the expression:
$E = 40^{\circ} + 60^{\circ} - 180^{\circ} = 100^{\circ} - 180^{\circ} = -80^{\circ}$.
Convert $-80^{\circ}$ to radians:
$-80 \times \dfrac{\pi}{180} = -\dfrac{8\pi}{18} = -\dfrac{4\pi}{9}$.
Answer: $-\dfrac{4 \pi}{9}$
First, simplify the angles:
$\cos(-680^{\circ}) = \cos(680^{\circ}) = \cos(720^{\circ} - 40^{\circ}) = \cos(-40^{\circ}) = \cos(40^{\circ})$.
Since $40^{\circ} \in [0, 180^{\circ}]$, $\cos^{-1}(\cos 40^{\circ}) = 40^{\circ}$.
Next, $\sin(-600^{\circ}) = -\sin(600^{\circ}) = -\sin(720^{\circ} - 120^{\circ}) = -(-\sin 120^{\circ}) = \sin 120^{\circ} = \sin(180^{\circ} - 60^{\circ}) = \sin 60^{\circ}$.
Since $60^{\circ} \in [-\pi/2, \pi/2]$, $\sin^{-1}(\sin 60^{\circ}) = 60^{\circ}$.
Finally, $\sin 270^{\circ} = -1$.
$\cos^{-1}(-1) = 180^{\circ}$.
Substituting these values into the expression:
$E = 40^{\circ} + 60^{\circ} - 180^{\circ} = 100^{\circ} - 180^{\circ} = -80^{\circ}$.
Convert $-80^{\circ}$ to radians:
$-80 \times \dfrac{\pi}{180} = -\dfrac{8\pi}{18} = -\dfrac{4\pi}{9}$.
Answer: $-\dfrac{4 \pi}{9}$
Hence, the correct answer is option: 'C'.