Question

\(cos^{-1}(cos\frac{4π}{3})+sin^{-1}(sin\frac{4π}{3}) = \ldots\)

• A. \(\frac{4π}{3}\)
• B. \(\frac{8π}{3}\)
• C. \(\frac{π}{3}\)
• D. \(\frac{3π}{2}\)

Hint:

Express altitudes with area and use the cosine and sine rules.

Answer 1

The Correct Option is C

Step 1: Test with an equilateral triangle:
Take side 2, so $R = \frac{2}{\sqrt3}$ and each altitude is $\sqrt3$. Each $\cos = \frac12$.

Step 2: Evaluate:
Sum = $3\times\frac{1/2}{\sqrt3} = \frac{3}{2\sqrt3} = \frac{\sqrt3}{2}$. And $\frac1R = \frac{\sqrt3}{2}$. They agree. The other choices differ: $R = 1.155$, $R^2 = 1.333$ and $\frac{1}{R^2} = 0.75$, none equal to 0.866. So (B).

Final Answer:
$\frac{1}{R}$. \[ \boxed{\frac{1}{R}} \]