Question

How do you find the value of `sin((5pi)/3)` using the double angle identity?

Answer 1

`-sqrt3/2`

Explanation:

Use trig table and unit circle:
`sin ((5pi)/3) = sin ((2pi)/3 + pi) = - sin ((2pi)/3) = -sqrt3/2`
Use double angle identity:
`sin 2a = 2sin a.cos a`
`sin ((10pi)/3) = 2sin ((5pi)/3).cos ((5pi)/3)`
`sin ((5pi)/3) = (sin ((10pi)/3))/(2cos ((5pi)/3))`
Because:
`cos ((5pi)/3) = cos ((2pi)/3 + pi) = - cos ((2pi)/3) = 1/2`
`sin ((10pi)/3) = sin (pi/3 + 3pi) = - sin (pi/3) = - sqrt3/2`
Therefore:
`sin ((5pi)/3) = (- sqrt3/2)/1 = - sqrt3/2`