Questions: tan(sin^(-1)(4/7))

Solution Steps

To solve the expression \(\tan \left(\sin ^{-1}\left(\frac{4}{7}\right)\right)\), we need to follow these steps:

1. Recognize that \(\sin^{-1}\left(\frac{4}{7}\right)\) represents an angle \(\theta\) such that \(\sin(\theta) = \frac{4}{7}\).
2. Use the Pythagorean identity to find \(\cos(\theta)\).
3. Use the definition of \(\tan(\theta)\) which is \(\frac{\sin(\theta)}{\cos(\theta)}\).

Given that \(\theta = \sin^{-1}\left(\frac{4}{7}\right)\), we have: \[ \sin(\theta) = \frac{4}{7} \]

Using the Pythagorean identity: \[ \cos^2(\theta) = 1 - \sin^2(\theta) \] Substituting the value of \(\sin(\theta)\): \[ \cos^2(\theta) = 1 - \left(\frac{4}{7}\right)^2 = 1 - \frac{16}{49} = \frac{49 - 16}{49} = \frac{33}{49} \] Thus, we find: \[ \cos(\theta) = \sqrt{\frac{33}{49}} = \frac{\sqrt{33}}{7} \approx 0.8207 \]

Using the definition of tangent: \[ \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} = \frac{\frac{4}{7}}{\frac{\sqrt{33}}{7}} = \frac{4}{\sqrt{33}} \approx 0.6963 \]

Final Answer