NIMCET 2012 — Mathematics PYQ

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Question

NIMCET 2012 — Mathematics PYQ

NIMCET | Mathematics | 2012

If $\sin(\pi \cos \theta) = \cos(\pi \sin \theta)$ , then $\sin 2\theta$ is equal to:

Choose the correct answer:

Explanation

1. Align the Trigonometric Functions:

We use the complementary angle identity $\cos A = \sin\left(\frac{\pi}{2} \pm A\right)$ to rewrite the equation:

\sin(\pi \cos \theta) = \sin\left(\frac{\pi}{2} \pm \pi \sin \theta\right)

2. Equate the Angles:

\pi \cos \theta = \frac{\pi}{2} \pm \pi \sin \theta

Divide the entire equation by $\pi$ :

\cos \theta = \frac{1}{2} \pm \sin \theta

\cos \theta \mp \sin \theta = \frac{1}{2}

3. Square Both Sides:

To find $\sin 2\theta$ , we square the equation:

(\cos \theta \mp \sin \theta)^2 = \left(\frac{1}{2}\right)^2

\cos^2 \theta + \sin^2 \theta \mp 2 \sin \theta \cos \theta = \frac{1}{4}

4. Simplify using Identities:

We know that $\sin^2 \theta + \cos^2 \theta = 1$ and $2 \sin \theta \cos \theta = \sin 2\theta$ :

1 \mp \sin 2\theta = \frac{1}{4}

Rearrange to solve for $\sin 2\theta$ :

\mp \sin 2\theta = \frac{1}{4} - 1

\mp \sin 2\theta = -\frac{3}{4}

\sin 2\theta = \pm \frac{3}{4}

Conclusion:

The value of $\sin 2\theta$ is $\pm \frac{3}{4}$ .

Correct Option: (a)