At what point does the terminal side of the angle (5\pi )/(6) in standard position intersect the unit circle?
recall that
x = r cosθ
y = r sinθ
now just plug in your numbers.
To determine at what point the terminal side of the angle (5π/6) in standard position intersects the unit circle, we need to use the unit circle and trigonometric functions.
Step 1: Understand the angles in standard position.
Angles in standard position are measured counterclockwise from the positive x-axis. The starting position is the initial side, which is along the positive x-axis. The terminal side is the side that we are interested in, which rotates counterclockwise.
Step 2: Convert the angle to degrees (optional).
Since we're given the angle (5π/6) in radians, we can convert it to degrees if needed. To convert radians to degrees, we use the formula: degrees = (radians × 180) / π. Applying this formula, we find that (5π/6) radians is equivalent to (150°) degrees.
Step 3: Find the reference angle.
The reference angle is the acute angle formed between the terminal side and the x-axis. To find the reference angle for an angle greater than 90°, we subtract the angle from 180°. In this case, the reference angle is 180° - 150° = 30°.
Step 4: Use the reference angle to determine the coordinates.
Since the terminal side intersects the unit circle, its coordinates on the unit circle will correspond to the trigonometric ratios of the reference angle.
For the reference angle of 30°, we know that it lies in the first quadrant of the unit circle. In the first quadrant, the x coordinate is positive, and the y coordinate is also positive. The main trigonometric functions that define the coordinates on the unit circle are sine (sin) and cosine (cos).
The sine of the reference angle (30°) is equal to the y-coordinate of the point on the unit circle. Therefore, sin(30°) = y-coordinate. Using a calculator or trigonometric tables, we find that sin(30°) = 0.5.
Similarly, the cosine of the reference angle is equal to the x-coordinate of the point on the unit circle. Therefore, cos(30°) = x-coordinate. Using a calculator or trigonometric tables, we find that cos(30°) = √3/2 or approximately 0.866.
Therefore, the point at which the terminal side of the angle (5π/6) intersects the unit circle is approximately (0.866, 0.5) or (cos(30°), sin(30°)).