Question

Which point forms a terminal side with the origin that creates a sine of LaTeX: -\frac{\sqrt{3}}{2} − √ 3 2 and a cosine of LaTeX: \frac{1}{2} 1 2 ?

Answer 1

To find the point that creates a sine of -√3/2 and a cosine of 1/2, look at the coordinates on the unit circle; the point (1/2, -√3/2) in the fourth quadrant meets these conditions.

Step-by-step explanation:

The student is asking how to find the point whose terminal side in standard position intersects the unit circle, thus producing a sine of -√3/2 and a cosine of 1/2.

We know from trigonometry that the sine of an angle is the y-coordinate of the corresponding point on the unit circle, and the cosine is the x-coordinate. Since sine is negative and cosine is positive, we are dealing with a point in the fourth quadrant. The coordinates we are looking for is the point at which the terminal side intersects the unit circle. Thus, the point with the coordinates (1/2, -√3/2) creates the desired sine and cosine values.

Answer 2

The point on the unit circle with a sine of -√3/2 and a cosine of 1/2 is (-1/2, -√3/2), located in the third quadrant.

Step-by-step explanation:

The student is asking about the coordinates of a point on the unit circle where the sine and cosine values are known. Since the cosine of an angle is represented by the adjacent side over the hypotenuse, cosine of 1/2 is associated with an angle whose adjacent side (x-coordinate) is 1/2 when the hypotenuse is 1.

Similarly, the sine, which is the opposite side over the hypotenuse, being -√3/2 means the opposite side (y-coordinate) is -√3/2.

These values correspond to an angle in the third quadrant of the unit circle, where both x and y are negative.

Hence, the point in question is (-1/2, -√3/2).