In an acute angled ΔABC, the least value of sec A + sec B + sec C is:
Explanation
Concept:
• If A + B + C = π, then cos A + cos B + cos C ≤ 3/2.
• AM-HM Inequality: AM ≥ HM.
For three numbers a, b and c:
⇒ a+b+c/3 ≥ 3/1/a+1/b+1/c
Calculation:
For the three angles of a ΔABC, A + B + C = π.
∴ cos A + cos B + cos C ≤ 3/2
⇒ cos A + cos B + cos C ≥ 2/3 ... (1)
Using the AM-HM inequality:
sec A + sec B + sec C ≥ 3/1/sec A + 1/sec B + 1/sec C
⇒ sec A + sec B + sec C ≥ 9/cos A + cos B + cos C
Using equation (1), we get:
∴ sec A + sec B + sec C ≥ 6
Therefore, the least value of sec A + sec B + sec C is 6.
