Question

In \(△ABC\), with usual notation, if \(a = 13,b = 14,c = 15\), then the sum of the values of \(sin(\frac{A}{2})\) and \(sinA\) is....

• A. \(\frac{14}{5\sqrt{5}}\)
• B. \(\frac{\sqrt{5}+4}{5}\)
• C. \(\frac{5+\sqrt{5}}{5}\)
• D. \(\frac{2}{\sqrt{5}}+\frac{1}{2}\)

Hint:

Use the cosine rule for cos A, then half angle formula.

Answer 1

The Correct Option is B

Step 1: Use the semi-perimeter:
$s = 21$, area $= \sqrt{21 \times 8 \times 7 \times 6} = 84$.

Step 2: Angle formulas:
$\sin A = \frac{2\Delta}{bc} = \frac{168}{210} = \frac45$.
$\sin\frac A2 = \sqrt{\frac{(s-b)(s-c)}{bc}} = \sqrt{\frac{7 \times 6}{210}} = \frac{1}{\sqrt5}$.
Sum $= \frac{\sqrt5}{5} + \frac45 = \frac{\sqrt5 + 4}{5}$.

Final Answer:
The sum is $\frac{\sqrt{5} + 4}{5}$, option (B). \[ \boxed{\frac{\sqrt{5}+4}{5}} \]