In the adjoining figure, ΔABC ABC is right-angled at B and ∠A= 30°. If BC = 6 cm, find (i) AB, (ii) AC.
Since, in a right-angled triangle,
sin θ = Perpendicular / Hypotenuse,
and cos θ = Base / Hypotenuse,
where θ is the angle made between the hypotenuse and the base.
(i) ∴ In the given figure, sin 30° = BC/AC
⇒ 1/2 = 6/AC
⇒ AC = 6 × 2
⇒ AC = 12 cm
(ii) Now, In the given figure, cos 30° = AB/AC
⇒ √3/2 = AB/12
⇒ (√3/2) × 12 = AB
⇒ AB = 6√3 cm
Aliter: Since ABC is a right-angled triangle,
∴ (AB)2 +(BC)2 = (AC)2
∴ (AB)2 = (AC)2 - (BC)2
⇒ (AB)2 = 144 – 36 = 108
⇒ (AB) = √108 = 6√3
∴ AB = 6√3 cm