The Terminal Ray Of Angle $\beta$, Drawn In Standard Position, Passes Through The Point $(-5, 2 \sqrt{7})$. What Is The Value Of \$\cos \beta$[/tex\]?Enter Your Answer In The Box In Simplest Radical Form.$\cos

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Introduction

In trigonometry, the terminal ray of an angle is a line segment that extends from the vertex of the angle to a point on the unit circle. The unit circle is a circle with a radius of 1, centered at the origin of the coordinate plane. In this article, we will explore how to find the cosine value of an angle given its terminal ray and a point it passes through.

The Problem

The terminal ray of angle $\beta$, drawn in standard position, passes through the point $(-5, 2 \sqrt{7})$. We need to find the value of $\cos \beta$.

The Unit Circle and Trigonometric Ratios

The unit circle is a fundamental concept in trigonometry. It is a circle with a radius of 1, centered at the origin of the coordinate plane. The unit circle is used to define the trigonometric ratios of sine, cosine, and tangent.

The cosine of an angle is defined as the ratio of the length of the adjacent side to the length of the hypotenuse in a right triangle. In the context of the unit circle, the cosine of an angle is the x-coordinate of the point where the terminal ray intersects the unit circle.

Finding the Cosine Value

To find the cosine value of angle $\beta$, we need to find the x-coordinate of the point where the terminal ray intersects the unit circle. We can do this by using the distance formula to find the distance from the origin to the point $(-5, 2 \sqrt{7})$.

The distance formula is given by:

d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}

In this case, we have:

d=(−5−0)2+(27−0)2d = \sqrt{(-5 - 0)^2 + (2 \sqrt{7} - 0)^2}

Simplifying the expression, we get:

d=25+28d = \sqrt{25 + 28}

d=53d = \sqrt{53}

The distance from the origin to the point $(-5, 2 \sqrt{7})$ is $\sqrt{53}$.

The Cosine Value

Now that we have the distance from the origin to the point $(-5, 2 \sqrt{7})$, we can find the cosine value of angle $\beta$. The cosine value is the x-coordinate of the point where the terminal ray intersects the unit circle.

Since the terminal ray passes through the point $(-5, 2 \sqrt{7})$, the x-coordinate of this point is $-5$. Therefore, the cosine value of angle $\beta$ is:

cos⁡β=−553\cos \beta = -\frac{5}{\sqrt{53}}

Simplifying the Cosine Value

We can simplify the cosine value by rationalizing the denominator. To do this, we multiply the numerator and denominator by $\sqrt{53}$.

cos⁡β=−553⋅5353\cos \beta = -\frac{5}{\sqrt{53}} \cdot \frac{\sqrt{53}}{\sqrt{53}}

Simplifying the expression, we get:

cos⁡β=−55353\cos \beta = -\frac{5 \sqrt{53}}{53}

Conclusion

In this article, we explored how to find the cosine value of an angle given its terminal ray and a point it passes through. We used the distance formula to find the distance from the origin to the point $(-5, 2 \sqrt{7})$, and then found the cosine value of angle $\beta$ by using the x-coordinate of the point where the terminal ray intersects the unit circle. The cosine value of angle $\beta$ is $-\frac{5 \sqrt{53}}{53}$.

Final Answer

Introduction

In our previous article, we explored how to find the cosine value of an angle given its terminal ray and a point it passes through. We used the distance formula to find the distance from the origin to the point $(-5, 2 \sqrt{7})$, and then found the cosine value of angle $\beta$ by using the x-coordinate of the point where the terminal ray intersects the unit circle.

In this article, we will answer some frequently asked questions related to the terminal ray of angle $\beta$ and its cosine value.

Q: What is the terminal ray of an angle?

A: The terminal ray of an angle is a line segment that extends from the vertex of the angle to a point on the unit circle.

Q: What is the unit circle?

A: The unit circle is a circle with a radius of 1, centered at the origin of the coordinate plane. It is used to define the trigonometric ratios of sine, cosine, and tangent.

Q: How do you find the cosine value of an angle?

A: To find the cosine value of an angle, you need to find the x-coordinate of the point where the terminal ray intersects the unit circle. You can do this by using the distance formula to find the distance from the origin to the point where the terminal ray intersects the unit circle.

Q: What is the distance formula?

A: The distance formula is given by:

d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}

Q: How do you simplify the cosine value?

A: You can simplify the cosine value by rationalizing the denominator. To do this, you multiply the numerator and denominator by $\sqrt{53}$.

Q: What is the final answer for the cosine value of angle $\beta$?

A: The final answer for the cosine value of angle $\beta$ is $-\frac{5 \sqrt{53}}{53}$.

Q: Can you explain the concept of the terminal ray in more detail?

A: The terminal ray of an angle is a line segment that extends from the vertex of the angle to a point on the unit circle. It is used to define the trigonometric ratios of sine, cosine, and tangent. The terminal ray is an essential concept in trigonometry, and it is used to find the values of sine, cosine, and tangent for any given angle.

Q: How do you find the sine value of an angle?

A: To find the sine value of an angle, you need to find the y-coordinate of the point where the terminal ray intersects the unit circle. You can do this by using the distance formula to find the distance from the origin to the point where the terminal ray intersects the unit circle.

Q: Can you provide an example of finding the cosine value of an angle?

A: Let's say we want to find the cosine value of angle $\alpha$, and the terminal ray of angle $\alpha$ passes through the point $(3, 4)$. We can use the distance formula to find the distance from the origin to the point $(3, 4)$, and then find the cosine value of angle $\alpha$ by using the x-coordinate of the point where the terminal ray intersects the unit circle.

Conclusion

In this article, we answered some frequently asked questions related to the terminal ray of angle $\beta$ and its cosine value. We explained the concept of the terminal ray, the unit circle, and the distance formula, and provided examples of finding the cosine value of an angle.

Final Answer

The final answer is: −55353\boxed{-\frac{5 \sqrt{53}}{53}}