Final answer:
To find the exact value of sin(585°), we can subtract 360° until we get an angle within one period. Sin(585°) is equal to sin(225°), which is equal to sin(-45°). The exact value of sin(-45°) is -√2/2.
Step-by-step explanation:
The exact value of sin(585°) can be determined by using the periodicity of the sine function. The sine function has a period of 360°, which means that the values of sin(x) repeat every 360°. To find the exact value of sin(585°), we can subtract 360° from 585° until we get an angle within one period. In this case, 585° - 360° = 225°. So, sin(585°) is equal to sin(225°).
We can apply the symmetry property of the sine function, which states that sin(180° - x) = sin(x). In this case, sin(225°) is equal to sin(180° - 225°) = sin(-45°).
The exact value of sin(-45°) is -√2/2. Therefore, the correct answer is option D: sin(585°) = -√2/2.