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Mathematics

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.

Trigonometrical Ratios

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Answer

(i) Given,

⇒ sin (θ + 36°) = cos θ

⇒ sin (θ + 36°) = sin (90° - θ)

⇒ θ + 36° = 90° - θ

⇒ 2θ = 90° - 36°

⇒ 2θ = 54°

⇒ θ = 542\dfrac{54}{2}

⇒ θ = 27°.

Hence, θ = 27°.

(ii) Given,

⇒ sec 4θ = cosec (θ - 20°)

⇒ sec 4θ = sec [90° - (θ - 20°)]

⇒ 4θ = [90° - (θ - 20°)]

⇒ 4θ = 110° - θ

⇒ 5θ = 110°

⇒ θ = 1105\dfrac{110}{5}

⇒ θ = 22°.

Hence, θ = 22°.

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