Question:
Evaluate \(\sin \left[\tan^{-1} \tan\left(\frac{3\pi}{4}\right)\right]\).
Evaluate \(\sin \left[\tan^{-1} \tan\left(\frac{3\pi}{4}\right)\right]\).
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Always reduce the angle to its principal value branch before applying inverse operations.
For sine and tangent, remember the negative sign 'comes out' of the function.
For sine and tangent, remember the negative sign 'comes out' of the function.
Updated On: Sep 10, 2026
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Solution and Explanation
Concept:
• Principal Value Branch of \(\tan^{-1}x\) is \(\left( -\frac{\pi}{2}, \frac{\pi}{2} \right)\).
• Relation: \(\tan^{-1}(\tan \theta) = \theta\) only if \(\theta\) belongs to the principal value range.
• Symmetry: \(\tan(\pi - \theta) = -\tan \theta\).
Step 1: Evaluate the inner tangent function value
The angle given is \(\frac{3\pi}{4}\). Since \(\frac{3\pi}{4}\) does not lie in the principal value branch \(\left( -\frac{\pi}{2}, \frac{\pi}{2} \right)\), we simplify it: \[ \tan\left(\frac{3\pi}{4}\right) = \tan\left(\pi - \frac{\pi}{4}\right) \] Using the identity \(\tan(\pi - \theta) = -\tan \theta\): \[ \tan\left(\frac{3\pi}{4}\right) = -\tan\left(\frac{\pi}{4}\right) = -1 \]
Step 2: Find the value of the inverse tangent function
Now we find \(\tan^{-1}(-1)\). We look for an angle \(\theta \in \left( -\frac{\pi}{2}, \frac{\pi}{2} \right)\) such that \(\tan \theta = -1\). \[ \tan^{-1}(-1) = -\frac{\pi}{4} \]
Step 3: Calculate the final sine value
The expression becomes: \[ \sin\left[ -\frac{\pi}{4} \right] \] Using the property \(\sin(-\theta) = -\sin \theta\): \[ \sin\left( -\frac{\pi}{4} \right) = -\sin\left( \frac{\pi}{4} \right) \] \[ = -\frac{1}{\sqrt{2}} \]
• Principal Value Branch of \(\tan^{-1}x\) is \(\left( -\frac{\pi}{2}, \frac{\pi}{2} \right)\).
• Relation: \(\tan^{-1}(\tan \theta) = \theta\) only if \(\theta\) belongs to the principal value range.
• Symmetry: \(\tan(\pi - \theta) = -\tan \theta\).
Step 1: Evaluate the inner tangent function value
The angle given is \(\frac{3\pi}{4}\). Since \(\frac{3\pi}{4}\) does not lie in the principal value branch \(\left( -\frac{\pi}{2}, \frac{\pi}{2} \right)\), we simplify it: \[ \tan\left(\frac{3\pi}{4}\right) = \tan\left(\pi - \frac{\pi}{4}\right) \] Using the identity \(\tan(\pi - \theta) = -\tan \theta\): \[ \tan\left(\frac{3\pi}{4}\right) = -\tan\left(\frac{\pi}{4}\right) = -1 \]
Step 2: Find the value of the inverse tangent function
Now we find \(\tan^{-1}(-1)\). We look for an angle \(\theta \in \left( -\frac{\pi}{2}, \frac{\pi}{2} \right)\) such that \(\tan \theta = -1\). \[ \tan^{-1}(-1) = -\frac{\pi}{4} \]
Step 3: Calculate the final sine value
The expression becomes: \[ \sin\left[ -\frac{\pi}{4} \right] \] Using the property \(\sin(-\theta) = -\sin \theta\): \[ \sin\left( -\frac{\pi}{4} \right) = -\sin\left( \frac{\pi}{4} \right) \] \[ = -\frac{1}{\sqrt{2}} \]
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