To simplify the expression (- sin(pi/12))/(sin((25 pi)/12)), we can start by using the fact that sin(-θ) = -sin(θ).
Given that sin(-pi/12) = -sin(pi/12), the expression can be rewritten as:
(-sin(pi/12))/(sin((37 pi)/12))
Now, let's focus on the denominator sin((37 pi)/12). To simplify it, we need to evaluate sin((37 pi)/12).
To find the value, we can use the periodicity property of sine function. The sine function repeats its values every 2Ï€ radians, so we can write:
sin((37 pi)/12) = sin((37 pi)/12 - 2Ï€)
Since 2Ï€ is equivalent to 24Ï€/12, we can simplify the expression further:
sin((37 pi)/12) = sin((37 pi)/12 - 24Ï€/12)
Now, let's simplify the angle inside the sin function:
(37 pi)/12 - 24Ï€/12 = (37 pi - 24Ï€)/12 = (13 pi)/12
Therefore, sin((37 pi)/12) is equivalent to sin((13 pi)/12).
Now, we have the simplified expression: (-sin(pi/12))/(sin((13 pi)/12))
By substituting these values, we have:
(-sin(pi/12))/(sin((13 pi)/12))
=(-sin(pi/12))/(sin((13 pi)/12))
Now, you can evaluate this expression using a calculator or a trigonometric identity for sin(pi/12). If you want to calculate it, remember to check if your calculator is in radians or degrees mode.