Sine Sum Formula Application Given and , find .

Trigonometry Solution – Problem 4: find

Sine Sum Formula Application

Given $\sin(2\alpha-\beta) = \frac{5}{13}$ and $\cos(\alpha-\beta) = \frac{1}{3}\sin\alpha$ , find $\sin\alpha$ .

We can write:

$\sin(2\alpha-\beta) = \sin[(\alpha-\beta)+\alpha]$

$= \sin(\alpha-\beta)\cos\alpha + \cos(\alpha-\beta)\sin\alpha$

$= \sin(\alpha-\beta)\cos\alpha + \frac{1}{3}\sin^2\alpha$

Therefore:

$\sin(\alpha-\beta)\cos\alpha = \frac{5}{13} - \frac{1}{3}\sin\alpha = \frac{1}{12}$

Thus:

$\sin\alpha = \frac{1}{3}\cdot\frac{1}{12} = \frac{1}{4}$

$\frac{1}{4}$

Correct identity setup (2 pts): choose an appropriate sum/difference, double-angle, or auxiliary-angle idea and set up the key equation(s).
Correct algebra / trig simplification (2 pts): transform expressions without sign mistakes.
Solve for target quantity (2 pts): isolate the requested value and handle any constraints if needed.
Final answer (1 pt): clearly state the result in the required form.