Question

In the xy-plane, the terminal ray of an angle in standard position intersects the unit circle at point p. which of the following is true about the cosine of the angle, as the angle measure increases from 0 to π

A the cosine of the angle increase, then decrease. because the vertical displacement of P form x-axis increase. then decrease over the interval
B the cosine of the angle decrease, then increase. because the vertical displacement of P form x-axis decrease .then increase over the interval
C the cosine of the angle decrease. because the vertical displacement of P form x-axis decrease. over the entire interval
D the cosine of the angle decrease. because the vertical displacement of P form y-axis decrease. over the entire interval

Answer 1

The cosine of an angle, which corresponds to the x-coordinate of point P on the unit circle, decreases continuously from 0 to π in the xy-plane as the angle measure increases. The correct description of this behavior is that the cosine decreases because the horizontal displacement of P from the y-axis decreases over this interval.

Step-by-step explanation:

In the xy-plane, as the angle measure in standard position increases from 0 to π, the terminal ray intersects the unit circle at various points. For an angle θ, the cosine of the angle corresponds to the x-coordinate of point P, the point of intersection on the unit circle. From 0 to π, the cosine starts at 1 (when the angle is 0 degrees and P is at (1,0)), decreases to 0 (when the angle is 90 degrees and P is at (0,1)), and continues decreasing until it reaches -1 (when the angle is 180 degrees and P is at (-1,0)). Therefore, the cosine of the angle in this interval decreases monotonically, and the correct choice that describes this behavior is:

Option C: The cosine of the angle decrease because the horizontal (not vertical) displacement of P from the y-axis (which corresponds to the cosine) decreases over the entire interval from 0 to π.

It's essential to note that while the cosine correlates to the x-coordinate of point P, the vertical displacement mentioned in the choices refers to the y-coordinate, which is related to the sine of the angle. In conclusion, the vertical displacement is unrelated to the decrease in cosine; instead, it's the horizontal displacement that causes the cosine value to decrease.

Answer 2

The cosine of the angle decreases, then increases as the angle measure increases from 0 to π.

Step-by-step explanation:

In the xy-plane, the terminal ray of an angle in standard position intersects the unit circle at point P. The cosine of the angle is equal to the x-coordinate of point P.

As the angle in standard position increases from 0 to π, point P moves clockwise along the unit circle. The x-coordinate (cosine) of P starts at 1 when the angle is 0, decreases to 0 when the angle is π/2, and decreases further to -1 when the angle is π.

Therefore, the correct answer is Option A: the cosine of the angle decrease, then increase. This is because the vertical displacement of point P from the x-axis decreases, then increases over the interval.