Find the value of θ if
(i) sin (θ + 36°) = cos θ, where θ and θ + 36° are acute angles.
(ii) sec 4θ = cosec (θ – 20°), where 4θ and θ – 20° are acute angles.
Answer :
(i) Given, θ and (θ + 36°) are acute angles
sin (θ + 36°) = cos θ = sin (90° – θ) [As, sin (90° – θ) = cos θ]
θ + 36° = 90° – θ
⇒ θ + θ = 90° – 36°
⇒ 2θ = 54°
⇒ θ = 54°/2
∴ θ = 27°
(ii) Given, θ and (θ – 20°) are acute angles
And,
sec 4θ = cosec (θ – 20°)
⇒ cosec (90° – 4θ) = cosec (θ – 20°) [Since, cosec (90° – θ) = sec θ]
On comparing, we get
90° – 4θ = θ – 20°
⇒ 90° + 20° = θ + 4θ
⇒ 5θ = 110°
⇒ θ = 110°/5
∴ θ = 22°
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