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

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Introduction

In mathematics, the terminal ray of an angle is the ray that extends from the vertex of the angle in the direction of the angle. The point P(x, y) is located on this terminal ray, and we are given that the angle θ is between π radians and 3π/2 radians. We are also given that the cosecant of θ is equal to -5/2. In this article, we will explore the relationship between the point P(x, y) and the angle θ, and we will use this information to find the coordinates of the point P.

Understanding the Angle θ

The angle θ is between π radians and 3π/2 radians, which means that it is in the third or fourth quadrant. Since the cosecant of θ is equal to -5/2, we can use this information to find the sine of θ.

Finding the Sine of θ

The cosecant of an angle is equal to the reciprocal of the sine of the angle. Therefore, we can write:

csc⁡θ=1sin⁡θ\csc \theta = \frac{1}{\sin \theta}

Since we are given that the cosecant of θ is equal to -5/2, we can set up the equation:

1sin⁡θ=−52\frac{1}{\sin \theta} = -\frac{5}{2}

To solve for the sine of θ, we can take the reciprocal of both sides of the equation:

sin⁡θ=−25\sin \theta = -\frac{2}{5}

Finding the Cosine of θ

Since the angle θ is in the third or fourth quadrant, the cosine of θ is negative. We can use the Pythagorean identity to find the cosine of θ:

cos⁡2θ+sin⁡2θ=1\cos^2 \theta + \sin^2 \theta = 1

Substituting the value of the sine of θ that we found earlier, we get:

cos⁡2θ+(−25)2=1\cos^2 \theta + \left(-\frac{2}{5}\right)^2 = 1

Simplifying the equation, we get:

cos⁡2θ+425=1\cos^2 \theta + \frac{4}{25} = 1

Subtracting 4/25 from both sides of the equation, we get:

cos⁡2θ=2125\cos^2 \theta = \frac{21}{25}

Taking the square root of both sides of the equation, we get:

cos⁡θ=−2125\cos \theta = -\sqrt{\frac{21}{25}}

Finding the Coordinates of P(x, y)

Since the point P(x, y) is located on the terminal ray of the angle θ, we can use the coordinates of the point to find the values of x and y. We can write:

x=rcos⁡θx = r \cos \theta

y=rsin⁡θy = r \sin \theta

where r is the distance from the origin to the point P.

Finding the Distance r

Since the point P(x, y) is located on the terminal ray of the angle θ, we can use the distance formula to find the distance r:

r=x2+y2r = \sqrt{x^2 + y^2}

Substituting the values of x and y that we found earlier, we get:

r=(−2125)2+(−25)2r = \sqrt{\left(-\sqrt{\frac{21}{25}}\right)^2 + \left(-\frac{2}{5}\right)^2}

Simplifying the equation, we get:

r=2125+425r = \sqrt{\frac{21}{25} + \frac{4}{25}}

r=2525r = \sqrt{\frac{25}{25}}

r=1r = 1

Finding the Coordinates of P(x, y)

Now that we have found the distance r, we can use the coordinates of the point to find the values of x and y. We can write:

x=rcos⁡θx = r \cos \theta

y=rsin⁡θy = r \sin \theta

Substituting the values of r, cos θ, and sin θ that we found earlier, we get:

x=1⋅(−2125)x = 1 \cdot \left(-\sqrt{\frac{21}{25}}\right)

y=1⋅(−25)y = 1 \cdot \left(-\frac{2}{5}\right)

Simplifying the equations, we get:

x=−2125x = -\sqrt{\frac{21}{25}}

y=−25y = -\frac{2}{5}

Conclusion

In this article, we explored the relationship between the point P(x, y) and the angle θ. We used the given information to find the sine and cosine of θ, and we used these values to find the coordinates of the point P. We found that the coordinates of the point P are (-√21/25, -2/5).

Final Answer

The final answer is: (−2125,−25)\boxed{(-\sqrt{\frac{21}{25}}, -\frac{2}{5})}

Introduction

In our previous article, we explored the relationship between the point P(x, y) and the angle θ. We used the given information to find the sine and cosine of θ, and we used these values to find the coordinates of the point P. In this article, we will answer some of the most frequently asked questions about the point P(x, y) on the terminal ray of angle θ.

Q: What is the terminal ray of an angle?

A: The terminal ray of an angle is the ray that extends from the vertex of the angle in the direction of the angle.

Q: What is the relationship between the point P(x, y) and the angle θ?

A: The point P(x, y) is located on the terminal ray of the angle θ. This means that the coordinates of the point P are related to the values of the sine and cosine of θ.

Q: How do you find the sine and cosine of θ?

A: To find the sine and cosine of θ, you can use the given information and the Pythagorean identity. The Pythagorean identity states that the sum of the squares of the sine and cosine of an angle is equal to 1.

Q: What is the distance r from the origin to the point P?

A: The distance r from the origin to the point P is equal to the square root of the sum of the squares of the x and y coordinates of the point P.

Q: How do you find the coordinates of the point P?

A: To find the coordinates of the point P, you can use the values of the sine and cosine of θ and the distance r from the origin to the point P.

Q: What are the coordinates of the point P?

A: The coordinates of the point P are (-√21/25, -2/5).

Q: What is the significance of the angle θ being between π radians and 3π/2 radians?

A: The angle θ being between π radians and 3π/2 radians means that the point P is located in the third or fourth quadrant.

Q: How do you use the given information to find the sine and cosine of θ?

A: To find the sine and cosine of θ, you can use the given information and the Pythagorean identity. The Pythagorean identity states that the sum of the squares of the sine and cosine of an angle is equal to 1.

Q: What is the relationship between the cosecant of θ and the sine of θ?

A: The cosecant of θ is equal to the reciprocal of the sine of θ.

Q: How do you find the cosecant of θ?

A: To find the cosecant of θ, you can use the given information and the reciprocal of the sine of θ.

Q: What is the significance of the cosecant of θ being equal to -5/2?

A: The cosecant of θ being equal to -5/2 means that the sine of θ is equal to -2/5.

Q: How do you use the given information to find the coordinates of the point P?

A: To find the coordinates of the point P, you can use the values of the sine and cosine of θ and the distance r from the origin to the point P.

Q: What are the final coordinates of the point P?

A: The final coordinates of the point P are (-√21/25, -2/5).

Conclusion

In this article, we answered some of the most frequently asked questions about the point P(x, y) on the terminal ray of angle θ. We hope that this article has been helpful in understanding the relationship between the point P and the angle θ.

Final Answer

The final answer is: (−2125,−25)\boxed{(-\sqrt{\frac{21}{25}}, -\frac{2}{5})}