Final answer:
The exact value of sin(345 degrees) is found by considering the reference angle of 15 degrees and applying the half-angle identity, yielding the final value of - √2/2.
Step-by-step explanation:
To find the exact value of sin(345 degrees), we can use the unit circle or reference angles. The reference angle for 345 degrees is 360 - 345 = 15 degrees. Since 345 degrees is in the fourth quadrant, where the sine values are negative, we take the sine of the positive reference angle and apply a negative sign.
Knowing that sin(15 degrees) is not one of the standard angles we memorize, we can instead use the half-angle identity sin(θ/2) = ± √½(1 - cos(θ)), where θ is the double of our reference angle, which is 30 degrees.
Therefore, sin(15) = √½(1 - cos(30))
Since cos(30 degrees) = √3/2, the calculation becomes sin(15) = √½(1 - √3/2), and we apply a negative sign to get the final value for sin(345 degrees):
sin(345 degrees) = - sin(15 degrees) = - √½(1 - √3/2) which simplifies to - √2/2.
Thus, the exact value for sin(345 degrees) is - √2/2, which corresponds to option (a).