Consider the following statements 1 If ABC is an equilateral triangle then 3 tan A B tan C 1 2 If AB...........

Mathematics (NDA) Trigonometry

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Consider the following statements

1. If ABC is an equilateral triangle, then 3 tan (A + B) tan C = 1.

2. If ABC is a triangle in which A = 78°, B = 66°, then tan(A/2 +C)

3. If ABC is any triangle, then

Which of the above statements is/are correct?

Only 1

Only 2

1 and 2

2 and 3

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Explanation:

1. Since, ABC is an equilateral triangle.
$\therefore$ $\angle$ A = $\angle$ B = $\angle$ C = 60°
Now, consider
3 tan (A + B) tan C = 3 tan 120° tan 60°
= 3 tan (180° − 60° ) tan 60°
= (−3) tan 60° tan 60° = (−3) × $\sqrt3$ × $\sqrt3$ = − 9
2. Consider, tan(A/2 +C)= tan(78°/2 +36°)
[ $\because$ $\angle$ C = 180° − $\angle$ A − $\angle$ B= 180° − 78° − 66° = 36°]
= tan 75° < tan 78°
[ $\because$ value of tan q increases, as q varying from 0° to 90°]
= tan A
Hence, tan(A/2 +C)