If sin\theta = (1)/(3) , (\pi )/(2) \ \textless \ \theta \ \textless \ \pi. Find the exact value of sin (\theta + (\pi )/(6))

Question

If
`sin\theta = (1)/(3)` ,
`(\pi )/(2) \ \textless \ \theta \ \textless \ \pi`. Find the exact value of


`sin (\theta + (\pi )/(6))`

Answer 1

- 0.183

Explanation:

Given that
`\sin \theta = (1)/(3)`

and
`(\pi )/(2) < \theta  < \pi`

We have to find the exact value of
`\sin (\theta + (\pi )/(6) )`.

Now,
`\sin \theta = (1)/(3)`

⇒
`\theta = \sin ^(-1) ((1)/(3) ) = 19.47`

Now, since
`(\pi )/(2) < \theta  < \pi`,

So,
`\theta  = 180 - 19.47 = 160.53`

{Since
`\sin \theta = \sin (180 - \theta)`

Now,
`\theta + (\pi )/(6) = 160.53 + 30 = 190.52`

Hence,
`\sin (\theta + (\pi )/(6) )`.

=
`\sin 190.52`

= - 0.183 (Approximate) (Answer)