Find sin β given sin α = 4/5 and cos(α+β) = -12/13 for angles in (0, π/2)

Trigonometric Problem: Find \(\sin \beta\)

Given that \(\alpha, \beta \in \left(0, \frac{\pi}{2}\right)\), \(\sin \alpha = \frac{4}{5}\) and \(\cos(\alpha+\beta) = -\frac{12}{13}\), determine \(\sin \beta\).

Select the correct option:

a. \(\frac{63}{65}\)

b. \(\frac{61}{65}\)

c. \(\frac{3}{5}\)

d. \(\frac{5}{13}\)

Detailed Solution

Step 1: Given \(\sin \alpha = \frac{4}{5}\) and \(\alpha \in \left(0, \frac{\pi}{2}\right)\), we have \[ \cos \alpha = \sqrt{1-\sin^2\alpha} = \sqrt{1 - \left(\frac{4}{5}\right)^2} = \frac{3}{5}. \]

Step 2: The cosine addition formula is \[ \cos(\alpha+\beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta. \] Substituting the known values: \[ \frac{3}{5}\cos \beta - \frac{4}{5}\sin \beta = -\frac{12}{13}. \] Multiplying both sides by 5: \[ 3\cos \beta - 4\sin \beta = -\frac{60}{13}. \]

Step 3: Notice that the coefficients \(3\) and \(-4\) correspond to a \(3-4-5\) triangle. Define an angle \(\phi\) such that \[ \cos \phi = \frac{3}{5} \quad \text{and} \quad \sin \phi = \frac{4}{5}. \] Then, \[ 3\cos \beta - 4\sin \beta = 5\left(\cos \phi \cos \beta - \sin \phi \sin \beta\right) = 5\cos(\beta+\phi). \] So the equation becomes: \[ 5\cos(\beta+\phi) = -\frac{60}{13} \quad \Rightarrow \quad \cos(\beta+\phi) = -\frac{12}{13}. \]

Step 4: Since \(\cos(\beta+\phi) = -\frac{12}{13}\), the angle \(\beta+\phi\) lies in the second quadrant (where cosine is negative). Its sine is: \[ \sin(\beta+\phi) = \sqrt{1-\cos^2(\beta+\phi)} = \sqrt{1-\left(-\frac{12}{13}\right)^2} = \frac{5}{13}. \]

Step 5: Now, using the sine subtraction formula to express \(\sin \beta\): \[ \sin \beta = \sin\left[(\beta+\phi)-\phi\right] = \sin(\beta+\phi)\cos\phi - \cos(\beta+\phi)\sin\phi. \] Substitute the known values: \[ \sin \beta = \left(\frac{5}{13}\right)\left(\frac{3}{5}\right) - \left(-\frac{12}{13}\right)\left(\frac{4}{5}\right). \] Simplify the expression: \[ \sin \beta = \frac{15}{65} + \frac{48}{65} = \frac{63}{65}. \]

Thus, the correct answer is \(\frac{63}{65}\), which corresponds to option a.