Find the slope of the radius of the unit circle that corresponds to the given angle.
7π/6 radians
the slope is, naturally, tan(7π/6)
what angle in rqadians correspond to 3 rotations around the unit cvircle
To find the slope of the radius of the unit circle corresponding to the angle 7π/6 radians, you can follow these steps:
1. Determine the coordinates of the point on the unit circle that corresponds to the given angle.
2. Use the coordinates to find the slope of the line that connects the origin of the unit circle (0,0) with the point on the unit circle.
Let's execute these steps:
Step 1: Determine the coordinates of the point on the unit circle.
The unit circle is centered at the origin (0,0). The angle 7π/6 radians is in the third quadrant of the unit circle. Starting from the positive x-axis and moving clockwise, the angle 7π/6 radians is 210 degrees. In the third quadrant, the x-coordinate of the point is negative, while the y-coordinate is positive.
The coordinates of the point on the unit circle can be found using the trigonometric functions cosine (cos) and sine (sin).
cos(7π/6) = -√3/2
sin(7π/6) = 1/2
Hence, the coordinates of the point on the unit circle are (-√3/2, 1/2).
Step 2: Find the slope of the line connecting the origin to the point on the unit circle.
The slope can be found using the formula:
slope = (change in y-coordinate) / (change in x-coordinate)
In this case, the change in y-coordinate is 1/2 and the change in x-coordinate is -√3/2.
slope = (1/2) / (-√3/2)
To simplify this, we multiply the numerator and denominator by the conjugate of -√3/2, which is √3/2:
slope = (1/2) * (2/√3) = 1/√3 = √3/3
Therefore, the slope of the radius of the unit circle corresponding to the angle 7π/6 radians is √3/3.