Question

How do you find the exact value of `sin((7pi)/6)-sin(pi/3)`?

Answer 1

Use trigonometric identities

Explanation:

Use that
`sin(x + pi) = -sin(x)`
to learn that
`sin((7pi)/6) - sin(pi/3) = sin(pi + pi/6) - sin(pi/3) = -sin(pi/6) - sin(pi/3)`.

Now look up in tables, or derive from the unit circle that
`sin(pi/6) = 1/2`
and
`sin(pi/3) = sqrt(3)/2`,
such that
`sin((7pi)/6) - sin(pi/3) = -sin(pi/6) - sin(pi/3) = -1/2 - sqrt(3)/2 = (-1-sqrt(3))/2`,

which is an exact value for the trigonometric expression.