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Question
The value of `sin pi/18 + sin pi/9 + sin (2pi)/9 + sin (5pi)/18` is given by ______.
Options
`sin (7pi)/18 + sin (4pi)/9`
1
`cos pi/6 + cos (3pi)/7`
`cos pi/9 + sin pi/9`
Solution
The value of `sin pi/18 + sin pi/9 + sin (2pi)/9 + sin (5pi)/18` is given by `sin (7pi)/18 + sin (4pi)/9`.
Explanation:
The given expression is `sin pi/18 + sin pi/9 + sin (2pi)/9 + sin (5pi)/18`
= `(sin (5pi)/18 + sin pi/18) + (sin (2pi)/9 + sin pi/9)`
= `2sin (((5pi)/18 + pi/18)/2) * cos (((5pi)/18 - pi/18)/2) + 2sin (((2pi)/9 + pi/9)/2)*cos(((2pi)/9 - pi/9)/2)`
= `2sin pi/6 * cos pi/9 + 2sin pi/6* cos pi/18`
= `2 xx 1/2 cos pi/9 + 2 xx 1/2 cos pi/18`
= `cos pi/9 + cos pi/18`
= `sin(pi/2 - pi/9) + sin(pi/2 - pi/18)`
= `sin (7pi)/18 + sin (8pi)/18`
= `sin (7pi)/18 + sin (4pi)/9`
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