Find the value of A if

Find the value of A if

(i) sin 3A = cos (A – 6°), where 3A and A – 6° are acute angles

(ii) tan 2A = cot (A – 18°), where 2A and A – 18° are acute angles

(iii) If sec 2A = cosec (A – 27°) where 2A is an acute angle, find the measure of ∠A.

Answer :

(i) sin 3A = cos (A – 6°), where 3A and A – 6° are acute angles

sin 3A = cos (A – 6°)

cos (90° – θ) = sin θ

cos (90° – 3A) = cos (A – 6°)

90° – 3A = A – 6°

90° + 6° = A + 3A

⇒ 96° = 4A

A = 96°/4 = 24°

Hence, the value of A is 24°.

(ii) tan 2A = cot (A – 18°)

cot (90° – θ) = tan θ

cot (90° – 2A) = cot (A – 18°)

90° – 2A = A – 18°

90° + 18° = A + 2A

3A = 108°

⇒ A = 108°/3 = 36°

Hence, the value of A is 36°.

(iii) sec 2A = cosec (A – 27°)

cosec (90° – θ) = sec θ

cosec (90° – 2A) = cos (A – 27°)

90° – 2A = A – 27°

90° + 27° = A + 2A

3A = 117°

⇒ A = 117°/3 = 39°

Hence, the value of A is 39°.

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