JEE Main 2022 · 30 Jul · Shift 2 · Mathematics Q90

← JEE Main 2022 · 30 Jul · Shift 2

MathematicsInteger+4 / -1Question 90 of 90

Let $S=\left\{\theta\in[0,2\pi]-\left\{\dfrac\pi2,\dfrac{3\pi}2\right\}:\sin^{-1}(\sin\theta)+\cos^{-1}(\cos\theta)+\tan^{-1}(\tan\theta)=\dfrac{4\pi}5\right\}$ . Then $\dfrac{30}\pi\displaystyle\sum_{\theta\in S}\theta$ is equal to _______.

Your answer

Answer and worked solution

Official answer: 68

By quadrant. On $[0,\frac\pi2)$ the sum is $3\theta$ : $\theta=\frac{4\pi}{15}$ . On $(\frac\pi2,\pi]$ it is $(\pi-\theta)+\theta+(\theta-\pi)=\theta$ : $\theta=\frac{4\pi}5$ . On $(\pi,\frac{3\pi}2)$ it is $2\pi-\theta$ : $\theta=\frac{6\pi}5$ . On $(\frac{3\pi}2,2\pi]$ it is $\theta-2\pi<0$ : none.

Total $\frac{30}\pi\cdot\frac{34\pi}{15}=68$ .

Answer: 68

Topics:Inverse trigonometric functions

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JEE Main 2022 · 30 Jul · Shift 2, Mathematics question 90