Question

In the following questions, two statements (i) and (ii) are given. Choose the valid statement.

(i) In ΔABC, `sin  (A + B)/2 = sin  C/2`

(ii) If sin 2A = cos (A – 6°) and 2A and (A – 6°) are acute then A = 35°

विकल्प

• Only (i)

• Only (ii)

• Both (i) and (ii)

• Neither (i) nor (ii)

Answer 1

Neither (i) nor (ii)

Explanation:

(i) A + B + C = 180°.

So, `(A + B)/2 = 90^circ - C/2`.

Hence, `sin ((A + B)/2)` 

= `sin ((90^circ - C)/2)`

= `cos (C/2)`, not `sin (C/2)`

Equality would hold only when C = 90°.

So, (i) is not generally true.

(ii) Rewrite cos (A – 6°)

= sin (90° – (A – 6°)) 

= sin (96° – A) 

So, sin 2A = sin (96° – A) gives 2A = 96° – A

⇒ A = 32°

or 2A = 180° – (96° – A) 

⇒ A = 84°

The extra condition that 2A and (A – 6°) are acute rules out A = 84°, leaving A = 32°, not 35°.

Thus, (ii) is false.