Sum of Squares Method Given and , find .

Trigonometry Solution – Problem 5: find

Sum of Squares Method

Given $\cos\alpha + \cos\beta = \frac{\sqrt{10}}{10}$ and $\sin\alpha + \sin\beta = \frac{3\sqrt{10}}{10}$ , find $\cos(\alpha-\beta)$ .

Squaring both equations:

$\cos^2\alpha + 2\cos\alpha\cos\beta + \cos^2\beta = \frac{1}{10} \quad (1)$

$\sin^2\alpha + 2\sin\alpha\sin\beta + \sin^2\beta = \frac{9}{10} \quad (2)$

Adding (1) and (2):

$1 + 2\cos\alpha\cos\beta + 2\sin\alpha\sin\beta + 1 = 1$

$2\cos(\alpha-\beta) = -1$

$\cos(\alpha-\beta) = -\frac{1}{2}$

$-\frac{1}{2}$

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.