To find the value of \(\sin^{-1}\dfrac{1}{\sqrt{5}} + \cos^{-1}\dfrac{3}{\sqrt{10}}\), we need to individually evaluate each inverse trigonometric term and then add them up.
- Recall the identity of inverse trigonometric functions:
- The relationship between sine and cosine involves the identity: \(\sin^{-1} x + \cos^{-1} x = \dfrac{\pi}{2}\) for \(-1 \leq x \leq 1\).
- We can apply this identity to simplify the expression as follows:
- Let us find the relationship between the two inverse trigonometric identities:
- Consider \(\theta = \sin^{-1}\dfrac{1}{\sqrt{5}}\), hence \(\sin\theta = \dfrac{1}{\sqrt{5}}\).
- For another angle phi (\(\phi\)), observe \(\phi = \cos^{-1}\dfrac{3}{\sqrt{10}}\), implying \(\cos\phi = \dfrac{3}{\sqrt{10}}\).
- To find the sum: \(\theta + \phi\):
- From the identity: \(\sin^{-1}x + \cos^{-1}x = \dfrac{\pi}{2}\).
- Equate as \(\theta + \phi = \dfrac{\pi}{2}\).
- Thus, \(\cos^{-1}\dfrac{3}{\sqrt{10}} = \dfrac{\pi}{2} - \sin^{-1}\dfrac{1}{\sqrt{5}}\).
- Adding both parts of the expression:
- \(\sin^{-1}\dfrac{1}{\sqrt{5}} + \cos^{-1}\dfrac{3}{\sqrt{10}} = \left(\sin^{-1}\dfrac{1}{\sqrt{5}} + \cos^{-1}\dfrac{3}{\sqrt{10}}\right)\) simplifies to just \(\dfrac{\pi}{2}\) as the identities cancel out the variable contributions to satisfy \(\dfrac{\pi}{2}\).
Therefore, \(\sin^{-1}\dfrac{1}{\sqrt{5}} + \cos^{-1}\dfrac{3}{\sqrt{10}} = \dfrac{\pi}{4}\), thus the correct answer is \(\pi/4\).
Hence, the answer is: \(\pi/4\).