The point $P(x, y)$ is on the terminal ray of angle $\theta$. If $\theta$ is between $\pi$ radians and $\frac{3 \pi}{2}$ radians and $\csc \theta=-\frac{5}{2}$, what are the coordinates of $P(x, y)$?
A. $P(-\sqrt{21},-2)$
B. $P(\sqrt{21},-2)$
C. $P(-2, \sqrt{21})$
D. $P(-2,-\sqrt{21})$
Answer (2)
Find sin θ from csc θ .
Use the fact that sin θ = r y to express y and r in terms of a constant k .
Use the equation x 2 + y 2 = r 2 to find x in terms of k , considering the quadrant of θ .
Determine the coordinates of P ( x , y ) and compare with the given options: P ( − 21 , − 2 ) .
Explanation
Analyze the given information The point P ( x , y ) lies on the terminal ray of angle θ . We are given that π < θ < 2 3 π , which means θ is in the third quadrant. In the third quadrant, both x and y coordinates are negative. We are also given that csc θ = − 2 5 .
Find the value of sin(theta) Since csc θ = − 2 5 , we have sin θ = c s c θ 1 = − 5 2 . We know that sin θ = r y , where r is the distance from the origin to the point P ( x , y ) , and r is always positive. Therefore, we can write y = − 2 k and r = 5 k for some positive constant k .
Calculate x We also know that x 2 + y 2 = r 2 . Substituting y = − 2 k and r = 5 k , we get x 2 + ( − 2 k ) 2 = ( 5 k ) 2 x 2 + 4 k 2 = 25 k 2 x 2 = 21 k 2 x = ± 21 k Since x is negative in the third quadrant, we have x = − 21 k .
Determine the coordinates of P(x, y) Therefore, the coordinates of the point P ( x , y ) are ( − 21 k , − 2 k ) . We need to find a value of k such that the coordinates match one of the given options. If k = 1 , then the coordinates are ( − 21 , − 2 ) .
Final Answer Comparing our result with the given options, we see that the coordinates of P ( x , y ) are ( − 21 , − 2 ) .
Examples
Understanding trigonometric functions and their values in different quadrants is crucial in various fields like physics and engineering. For instance, when analyzing the motion of a pendulum, the angle θ it makes with the vertical changes over time, and the position of the pendulum bob can be described using trigonometric functions. Knowing the quadrant in which the angle lies helps determine the signs of the coordinates, which in turn helps in predicting the pendulum's position and velocity.
The coordinates of the point P ( x , y ) on the terminal ray of angle θ are P ( − 21 , − 2 ) , which corresponds to option A.
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