Find the value of θ.

(i) If 2θ is an acute angle and 2 sin 2θ = √3, find the value of θ.

(ii) If 20° + x is an acute angle and cos (20° + x) = sin 60°, then find the value of x.

(iii) If 3 sin² θ = 2 ¼ and θ is less than 90°, find the value of θ.

Answer :

(i) 2θ is an acute angle

2 sin 2θ = √3

sin 2θ = √3/2 = sin 60°

2θ = 60°

θ = 60°/2 = 30°

Hence, θ = 30°

(ii) 20° + x is an acute angle

cos (20° + x) = sin 60°

cos (20° + x) = sin 60° = cos (90° – 60°)

= cos 30°

20° + x = 30°

⇒ x = 30° – 20° = 10°

Hence, x = 10⁰.

(iii) 3 sin² θ = 2 ¼

θ is less than 90°

sin² θ = 9/(4 × 3) = ¾

sin θ = √3/2 = sin 60°

θ = 60°

Hence, θ = 60°.

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