The point $P(x, y)$ lies on the terminal side of an angle $\theta=\frac{3 \pi}{4}$ in standard position. What are the signs of the values of $x$ and $y$?
A. Both $x$ and $y$ are positive.
B. Both $x$ and $y$ are negative.
C. $x$ is positive, and $y$ is negative.
D. $x$ is negative, and $y$ is positive.
Answer (2)
Determine that the angle θ = 4 3 π lies in the second quadrant.
Recall that in the second quadrant, x is negative and y is positive.
Conclude that for the point P ( x , y ) , x is negative and y is positive.
State the final answer: x is negative, and y is positive.
Explanation
Analyze the problem and the given angle. The angle θ = 4 3 π is in standard position, meaning its initial side is on the positive x-axis and it's measured counterclockwise. To determine the signs of the coordinates ( x , y ) of a point on the terminal side of this angle, we need to identify the quadrant in which the angle lies.
Determine the quadrant of the angle. Since 2 π < 4 3 π < π , the angle 4 3 π lies in the second quadrant.
Determine the signs of x and y in the second quadrant. In the second quadrant, the x -coordinate is negative, and the y -coordinate is positive. Therefore, for the point P ( x , y ) on the terminal side of the angle θ = 4 3 π , x is negative, and y is positive.
State the final answer. Therefore, the sign of x is negative, and the sign of y is positive.
Examples
Understanding the signs of coordinates in different quadrants is crucial in various real-world applications. For instance, when analyzing the motion of an object in a circular path, knowing the quadrant helps determine the direction of the object's velocity and acceleration components. Similarly, in navigation, the signs of coordinates are essential for determining the location and direction of travel. Consider a radar system tracking an airplane; if the airplane's angle is 4 3 π relative to the radar, we know it's in the northwest direction, meaning its east-west position (x-coordinate) is negative, and its north-south position (y-coordinate) is positive.
The point P ( x , y ) lies on the terminal side of the angle θ = 4 3 π , which is in the second quadrant, indicating that x is negative and y is positive. Thus, the answer is option D: x is negative, and y is positive.
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