In mathematics, logarithmic manipulation is a powerful tool to simplify expressions involving logs. In this problem, we will be using this tool to evaluate the value of an expression containing a series of logarithms.


Evaluate log sin1 + log sin2 + log sin3 + ... + log sin179.




Using the logarithmic identity log(a*b) = log(a) + log(b), we can simplify the given expression as:

log(sin1 * sin2 * sin3 * ... * sin179)


Now, we can use the product-to-sum identity to write the product of sines as a sum of sines. The identity is given by:

sin(a) * sin(b) = (1/2)[cos(a-b) - cos(a+b)]

Using this identity, we can rewrite the product of sines as:

sin1 * sin2 * sin3 * ... * sin179

= (1/2)[cos(1-2) - cos(1+2)] * (1/2)[cos(2-3) - cos(2+3)] * ... * (1/2)[cos(178-179) - cos(178+179)]

= (1/2) * [(cos1 - cos2) * (cos2 - cos3) * ... * (cos178 - cos179)]


Now, we can use the logarithmic identity log(a*b) = log(a) + log(b) to write the above expression as:

log(sin1 * sin2 * sin3 * ... * sin179) = log[(1/2) * (cos1 - cos2) * (cos2 - cos3) * ... * (cos178 - cos179)]

= log(1/2) + log(cos1 - cos2) + log(cos2 - cos3) + ... + log(cos178 - cos179)


We can evaluate each of the logarithms using the identity log(a-b) - log(a+b) = 2*log(sin((a-b)/2)):

log(cos1 - cos2) = log[2*sin((1+2)/2)*sin((1-2)/2)]

= log(sin(1/2)) - log(sin(3/2))

Similarly, we can evaluate all the other logarithms. Substituting the values in the expression, we get:

log(sin1 + log sin2 + log sin3 + ... + log sin179) = log(1/2) + log(sin(1/2)) - log(sin(3/2)) + log(sin(3/2)) - log(sin(5/2)) + ... + log(sin(177/2)) - log(sin(179/2))

= log(1/2) + log(sin(1/2)) - log(sin(179/2))

= log(1/2*sin(1/2)) - log(sin(179/2))

= log(sin(89