Educational

Principal Value of sin⁻¹{cos(sin⁻¹(√3/2))} Question: Find the principal value of: \[ \sin^{-1}\left\{\cos\left(\sin^{-1}\left(\frac{\sqrt{3}}{2}\right)\right)\right\} \] Solution: Step 1: Evaluate inner inverse \[ \sin^{-1}\left(\frac{\sqrt{3}}{2}\right) = \frac{\pi}{3} \] Step 2: Substitute \[ \sin^{-1}\left(\cos \frac{\pi}{3}\right) \] Step 3: Evaluate cosine \[ \cos \frac{\pi}{3} = \frac{1}{2} \] Step 4: Final inverse \[ \sin^{-1}\left(\frac{1}{2}\right) = \frac{\pi}{6} \] — Final Answer: \[ \boxed{\frac{\pi}{6}} […]

Principal Value sin⁻¹(1/2) – 2sin⁻¹(1/√2) Question: Find the principal value of: \[ \sin^{-1}\left(\frac{1}{2}\right) – 2\sin^{-1}\left(\frac{1}{\sqrt{2}}\right) \] Solution: Step 1: Use standard values \[ \sin^{-1}\left(\frac{1}{2}\right) = \frac{\pi}{6} \] \[ \sin^{-1}\left(\frac{1}{\sqrt{2}}\right) = \frac{\pi}{4} \] Step 2: Substitute \[ \frac{\pi}{6} – 2\left(\frac{\pi}{4}\right) \] Step 3: Simplify \[ = \frac{\pi}{6} – \frac{\pi}{2} \] \[ = \frac{\pi – 3\pi}{6} =

Principal Value of sin⁻¹(tan 5π/4) Question: Find the principal value of: \[ \sin^{-1}\left(\tan \frac{5\pi}{4}\right) \] Solution: Step 1: Evaluate tangent value \[ \tan \frac{5\pi}{4} = 1 \] Step 2: Substitute \[ \sin^{-1}(1) \] Step 3: Use standard value \[ \sin^{-1}(1) = \frac{\pi}{2} \] Step 4: Check principal range \[ \frac{\pi}{2} \in \left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \] —

Principal Value of sin⁻¹(cos 3π/4) Question: Find the principal value of: \[ \sin^{-1}\left(\cos \frac{3\pi}{4}\right) \] Solution: Step 1: Evaluate cosine value \[ \cos \frac{3\pi}{4} = -\frac{\sqrt{2}}{2} \] Step 2: Substitute \[ \sin^{-1}\left(-\frac{\sqrt{2}}{2}\right) \] Step 3: Use standard value \[ \sin\left(-\frac{\pi}{4}\right) = -\frac{\sqrt{2}}{2} \] Step 4: Check principal range \[ -\frac{\pi}{4} \in \left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \] —

Principal Value of sin⁻¹((√3+1)/(2√2)) Question: Find the principal value of: \[ \sin^{-1}\left(\frac{\sqrt{3}+1}{2\sqrt{2}}\right) \] Solution: Step 1: Use identity Recall: \[ \sin\left(\frac{5\pi}{12}\right) = \frac{\sqrt{6} + \sqrt{2}}{4} \] Now simplify: \[ \frac{\sqrt{3}+1}{2\sqrt{2}} = \frac{\sqrt{6} + \sqrt{2}}{4} \] Step 2: Substitute \[ \sin^{-1}\left(\frac{\sqrt{6} + \sqrt{2}}{4}\right) \] Step 3: Check principal range \[ \frac{5\pi}{12} \notin \left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \] Use

Principal Value of sin⁻¹((√3−1)/(2√2)) Question: Find the principal value of: \[ \sin^{-1}\left(\frac{\sqrt{3}-1}{2\sqrt{2}}\right) \] Solution: Step 1: Use identity Recall: \[ \sin\left(\frac{\pi}{12}\right) = \frac{\sqrt{6} – \sqrt{2}}{4} \] Now simplify given expression: \[ \frac{\sqrt{3}-1}{2\sqrt{2}} = \frac{\sqrt{6} – \sqrt{2}}{4} \] Step 2: Substitute \[ \sin^{-1}\left(\frac{\sqrt{6} – \sqrt{2}}{4}\right) \] Step 3: Recognize value \[ = \frac{\pi}{12} \] Step 4:

Principal Value of sin⁻¹(cos 2π/3) Question: Find the principal value of: \[ \sin^{-1}\left(\cos \frac{2\pi}{3}\right) \] Solution: Step 1: Evaluate cosine value \[ \cos \frac{2\pi}{3} = -\frac{1}{2} \] Step 2: Substitute \[ \sin^{-1}\left(-\frac{1}{2}\right) \] Step 3: Use standard value \[ \sin\left(-\frac{\pi}{6}\right) = -\frac{1}{2} \] Step 4: Check principal range \[ -\frac{\pi}{6} \in \left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \] —

Principal Value of sin⁻¹(-√3/2) Question: Find the principal value of: \[ \sin^{-1}\left(-\frac{\sqrt{3}}{2}\right) \] Concept: The principal value range of \( \sin^{-1}x \) is: \[ -\frac{\pi}{2} \leq y \leq \frac{\pi}{2} \] — Solution: Step 1: Recall standard value \[ \sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2} \] Step 2: Apply negative sign Since sine is negative, the angle lies in

Number of Commutative Binary Operations on 2 Elements Question: The number of commutative binary operations that can be defined on a set of 2 elements is: (a) 8 (b) 6 (c) 4 (d) 2 Concept: For a set with \( n \) elements: Total binary operations = \( n^{n^2} \) Commutative operations depend only on

Number of Binary Operations on Set of 2 Elements Question: The number of binary operations that can be defined on a set with 2 elements is: (a) 8 (b) 4 (c) 16 (d) 64 Concept: Number of binary operations on a set with \( n \) elements is: \[ n^{n^2} \] Solution: Here \( n