Next, we will find the value of cos900°900°900°. Since cos900°900°900° = cos900°−360°900° - 360°900°−360° = cos540°540°540° = cos180°180°180° = -1.
Finally, to find ctg675°675°675°, we use the relationship between cotangent and tangent: ctg675°675°675° = 1/tan675°675°675° = 1/tan675°−360°675° - 360°675°−360° = 1/tan315°315°315°
In the fourth quadrant, the tangent function is positive, so we find the reference angle for 315°: 315° - 270° = 45°
This means that tan315°315°315° = tan45°45°45° = 1, and therefore ctg675°675°675° = 1.
Putting it all together: -810° + cos 900° - ctg 675° = -1 + −1-1−1 - 1 = -3
To find the value of sin−810°-810°−810°, we must first find the equivalent angle between 0° and 360°.
-810° + 360° = -450°
sin−450°-450°−450° = sin360°−450°360° - 450°360°−450° = sin−90°-90°−90° = -sin90°90°90° = -1
Next, we will find the value of cos900°900°900°. Since cos900°900°900° = cos900°−360°900° - 360°900°−360° = cos540°540°540° = cos180°180°180° = -1.
Finally, to find ctg675°675°675°, we use the relationship between cotangent and tangent:
ctg675°675°675° = 1/tan675°675°675° = 1/tan675°−360°675° - 360°675°−360° = 1/tan315°315°315°
In the fourth quadrant, the tangent function is positive, so we find the reference angle for 315°:
315° - 270° = 45°
This means that tan315°315°315° = tan45°45°45° = 1, and therefore ctg675°675°675° = 1.
Putting it all together:
-810° + cos 900° - ctg 675° = -1 + −1-1−1 - 1 = -3
Therefore, the final answer is -3.