Final answer:
The coordinates of the terminal point determined by t=(10\pi)/3 will be in the fourth quadrant, where the x-coordinate is positive and the y-coordinate is negative, denoted as (+, −). Option C is correct.
Step-by-step explanation:
To find the coordinates of the terminal point determined by t=(10\pi)/3, we first recognize that this is a problem of trigonometry involving the unit circle. When we see an angle like t=(10\pi)/3, it is measured in radians and represents the angle from the positive x-axis to the radius of the unit circle that forms this angle in the counterclockwise direction. The terminal point is where this radius intersects the unit circle.
To find the quadrant of the terminal point, we can divide t by 2\pi to find how many full rotations plus some extra angle are there. The angle (10\pi)/3 is equivalent to 3(2\pi)/3 + (4\pi)/3, indicating a full three rotations plus an additional (4\pi)/3 radians. Since 2\pi radians is a full circle, the extra (4\pi)/3 falls into the fourth quadrant because \pi to 3\pi/2 radians covers the third quadrant, and thus (4\pi)/3 is beyond 3\pi/2.
In the fourth quadrant, the x-coordinate (horizontally to the right side of the coordinate system) is positive, and the y-coordinate (vertically downward in the coordinate system) is negative. Therefore, the correct answer is the third option, which is (+, −), corresponding to a point in the fourth quadrant with a positive x-value and a negative y-value.