Do you know where is this angle?

Geometry Level 1

If sin ⁡ A = 3 5 \sin A = \dfrac 35 and 0 ∘ < A < 9 0 ∘ 0^\circ < A < 90^\circ , find sin ⁡ 4 A \sin 4A .

336 25 \frac{336}{25} 48 25 \frac{48}{25} 48 625 \frac{48}{625} 336 625 \frac{336}{625}

César Castro by César Castro , 20, USA

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1 solution

Chew-Seong Cheong
Apr 18, 2018

sin ⁡ A = 3 5 Given ⟹ cos ⁡ A = 1 − sin ⁡ 2 A = 1 − 9 25 = 4 5 sin ⁡ 2 A = 2 sin ⁡ A cos ⁡ A = 2 × 3 5 × 4 5 = 24 25 cos ⁡ 2 A = 1 − sin ⁡ 2 2 A = 1 − 576 625 = 7 25 ⟹ sin ⁡ 4 A = 2 sin ⁡ 2 A cos ⁡ 2 A = 2 × 24 25 × 7 25 = 336 625 \begin{aligned} \sin A & = \frac 35 & \small \color{#3D99F6} \text{Given} \\ \implies \cos A & = \sqrt{1-\sin^2 A} = \sqrt{1-\frac 9{25}} = \frac 45 \\ \sin 2A & = 2\sin A\cos A = 2\times \frac 35 \times \frac 45 = \frac {24}{25} \\ \cos 2A & = \sqrt{1-\sin^2 2A} = \sqrt{1-\frac {576}{625}} = \frac 7{25} \\ \implies \sin 4A & = 2\sin 2A\cos 2A = 2 \times \frac {24}{25}\times \frac 7{25} = \boxed{\dfrac {336}{625}} \end{aligned}

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