Question

Which of the following trigonometric ratios is equivalent to sin 4pi / 3?

cos(5pi/6)
cos(5pi/3)
sin(pi/3)
sin(7pi/6)

Answer 1

(A)

Explanation:

The given trigonometric ratio is :

`sin\frac{4{\pi}}{3}`

On solving this, we get

`sin\frac{4{\pi}}{3}`

=
`sin240^{{\circ}}`

=
`sin(180+60)`

=
`-sin60^{{\circ}}`

=
`-(√(3))/(2)`

(A) The given trigonometric ratio is :

`cos\frac{5{\pi}}{6}`

=
`cos150^{{\circ}}`

=
`cos(180-30)`

=
`-cos30^{{\circ}}`

=
`-(√(3))/(2)`

Which is equivalent to the given trigonometric ratio, thus (A) is correct.

(B) The given trigonometric ratio is :

`cos\frac{5{\pi}}{3}`

=
`cos300^{{\circ}}`

=
`cos(360-60)`

=
`-cos60^{{\circ}}`

=
`-(1)/(2)`

which is not equivalent to the given trigonometric ratio, thus(B) is incorrect.

(C) The given trigonometric ratio is :

`sin\frac{{\pi}}{3}`

=
`(√(3))/(2)`

which is not equivalent to the given trigonometric ratio, thus(C) is incorrect.

(D) The given trigonometric ratio is :

`sin\frac{7{\pi}}{6}`

=
`sin210^{{\circ}}`

=
`sin(180+30)`

=
`-sin30^{{\circ}}`

=
`-(1)/(2)`

which is not equivalent to the given trigonometric ratio, thus(D) is incorrect.