Determine the value of \(\sin \frac{{21\pi }}{2}\) and cos(-1740°)?

Question

Question

This question was previously asked in

AAI ATC Junior Executive 25 March 2021 Official Paper (Shift 1)

Answer 1

Concept:

cos (-θ) = cos θ

cos (2nπ - θ) = cos θ

Calculation:

We have,

⇒ \(sin(\frac{21π}{2})\)

Remove full rotations of 2π until the angle is between 0 and 2π

⇒ \(sin(\frac{21π}{2})= sin(5\times 2\pi +\frac{π}{2}) =sin(\frac{π}{2})\)

As we know that the value of \(sin(\frac{π}{2})\) is 1.

As we know, the value of cos θ repeats after an interval of 2π or 360°

⇒ cos (-1740°) = cos (1740°) [∵ cos (-θ) = cos θ]

We know that cos (2nπ - θ) = cos θ

As sin θ and cos θ are have a time period of 360°

⇒ cos (1740°) = cos (5 × 360° - 1740°)

⇒ cos (1740°) = cos (1800° - 1740°)

⇒ cos (1740°) = cos (60°)

As we know, cos 60° = 1/2

⇒ cos (-1740°) = 1/2.

∴ The value of \(\sin \frac{{21\pi }}{2}\) is 1 and the value of cos (-1740°) is 1/2.