Identify the terminal point for a 30° angle in a unit circle.
A (1/2, (sqrt(3))/2)
B. (1/3, (sqrt(3))/3)
c. ((sqrt(3))/2, 1/2)
D. ((sqrt(2))/2, (sqrt(2))/2)
The Correct Answer and Explanation is :
The correct answer is A. (1/2, √3/2).
Explanation:
A unit circle is a circle with a radius of 1 unit, centered at the origin (0,0) on the coordinate plane. Any point on the circumference of the unit circle corresponds to an angle measured from the positive x-axis, and the coordinates of that point are derived from the trigonometric functions cosine (cos) and sine (sin).
For an angle of 30°, we calculate the coordinates of the terminal point using these functions:
- The x-coordinate is the cosine of the angle.
- The y-coordinate is the sine of the angle.
- Cos(30°): Cosine of 30° is the adjacent side over the hypotenuse in a 30-60-90 triangle. For this triangle, the adjacent side (x-coordinate) is √3/2. However, this is a common mistake. In the unit circle, for 30°:
[
\cos(30°) = \frac{\sqrt{3}}{2}
] - Sin(30°): Sine of 30° is the opposite side over the hypotenuse in the same 30-60-90 triangle. In this case, the opposite side (y-coordinate) is:
[
\sin(30°) = \frac{1}{2}
]
Thus, the coordinates for a 30° angle in the unit circle are (cos(30°), sin(30°)) = (√3/2, 1/2).
However, notice that this matches option C, not A.
Upon further reflection, C should actually be the correct answer (√3/2, 1/2). This may be an error in the provided choices,