Trigonometry
Trigonometry
Allen Star Batch
Grade 11
Question:
If $\sin a + \sin \beta = \frac{3\sqrt{2}}{5}$ and $\cos a + \cos \beta = \frac{4\sqrt{2}}{5}$, then :
$\sin(a + \beta) = \frac{12}{13}$
$\sin(a + \beta) = \frac{24}{25}$
$\cos(a + \beta) = \frac{5}{13}$
$\cos(a + \beta) = \frac{7}{25}$
Step-by-Step Solution
Key Concept: Sum-to-product formulas combined with division allow us to find half-angle tangent, which then determines the full angle.
Given $\sin a + \sin b = 2\sin\frac{a+b}{2}\cos\frac{a-b}{2} = \frac{3\sqrt{2}}{5}$ and $\cos a + \cos b = 2\cos\frac{a+b}{2}\cos\frac{a-b}{2} = \frac{4\sqrt{2}}{5}$, dividing yields $\tan\frac{a+b}{2} = \frac{3}{4}$. Using the half-angle formula: $\sin(a+b) = \frac{24}{25}$ and $\cos(a+b) = \frac{7}{25}$.
Correct Answer: 2,4