Point P is located at the intersection of the unit circle and the terminal side of angle theta in standard position. Find the coordinates of P to the nearest thousandth.

theta = -105degrees

Please explain the steps. I'm completely stuck.

Θ is 105º clockwise from the x-axis

this gives a 75º reference angle with the negative x-axis

the x and y coordinates are both negative

x = - cos(75º)

y = - sin(75º)

To find the coordinates of point P on the unit circle, we can use the trigonometric ratios sine and cosine. By using the angle theta and the unit circle, we can determine the x and y coordinates of point P.

Here are the steps to find the coordinates of point P:
1. Draw the unit circle with its center at the origin (0,0) on a coordinate plane.
2. Place the angle theta in standard position, which means starting from the positive x-axis and rotating in the counterclockwise direction.
3. The given angle theta is -105 degrees, which means it is in the fourth quadrant since it is negative. Recall that angles in the fourth quadrant have positive x-coordinates and negative y-coordinates.
4. To determine the x-coordinate of point P, use the cosine function. The cosine of an angle is defined as the adjacent side (x-coordinate) divided by the hypotenuse (which is always 1 on the unit circle).
In this case, the cosine of -105 degrees is equal to the x-coordinate of point P.
cos(-105°) = x-coordinate of point P
5. Use a scientific calculator set to degree mode to find the cosine value. Enter -105 degrees and find its cosine value. Round the result to the nearest thousandth.
6. To determine the y-coordinate of point P, use the sine function. The sine of an angle is defined as the opposite side (y-coordinate) divided by the hypotenuse (which is always 1 on the unit circle).
In this case, the sine of -105 degrees is equal to the y-coordinate of point P.
sin(-105°) = y-coordinate of point P
7. Use a scientific calculator set to degree mode to find the sine value. Enter -105 degrees and find its sine value. Round the result to the nearest thousandth.
8. The x-coordinate and y-coordinate rounded to the nearest thousandth are the coordinates of point P on the unit circle.

Therefore, the coordinates of point P to the nearest thousandth are (x, y), where x is the cosine value rounded, and y is the sine value rounded for the given angle theta of -105 degrees.

To find the coordinates of point P, we will use the unit circle and the angle theta in standard position.

Step 1: Draw the unit circle.
Start by drawing a circle with a radius of 1 unit. This is called the unit circle because the radius is 1.

Step 2: Locate the angle theta.
Locate the angle theta on the unit circle's terminal side. In this case, theta is -105 degrees, which means we will be rotating in the clockwise direction from the positive x-axis.

Step 3: Determine the reference angle.
The reference angle is the positive acute angle formed between the terminal side and the x-axis. Since theta is a negative angle, we consider the positive angle equivalent to it by taking the absolute value. Therefore, the reference angle for -105 degrees is 105 degrees.

Step 4: Identify the coordinates on the unit circle.
On the unit circle, the x-coordinate represents the cosine of the angle, and the y-coordinate represents the sine of the angle. Since we are rotating clockwise, we need to find the cosine and sine of the reference angle, which is 105 degrees.

Step 5: Calculate the cosine and sine of the reference angle.
Using a scientific calculator, calculate the cosine of 105 degrees and round the result to the nearest thousandth. Then, calculate the sine of 105 degrees and round the result to the nearest thousandth.

cos(105°) ≈ -0.259
sin(105°) ≈ 0.966

Step 6: Determine the coordinates of point P.
The coordinates of point P are (-0.259, 0.966) rounded to the nearest thousandth.

Therefore, the coordinates of point P to the nearest thousandth are (-0.259, 0.966).