The Terminal Side Of An Angle \[$\theta\$\] In Standard Position Intersects The Unit Circle At \[$\left(-\frac{\sqrt{17}}{8},-\frac{\sqrt{47}}{8}\right)\$\]. What Is \[$\cos (\theta)\$\]?

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Introduction

In trigonometry, the unit circle is a fundamental concept used to define the trigonometric functions of an angle. The terminal side of an angle in standard position intersects the unit circle at a point, and the coordinates of this point are used to determine the values of the trigonometric functions. In this article, we will explore how to find the cosine of an angle given the coordinates of the point where the terminal side intersects the unit circle.

The Unit Circle and Trigonometric Functions

The unit circle is a circle with a radius of 1 centered at the origin of the coordinate plane. The terminal side of an angle in standard position intersects the unit circle at a point, and the coordinates of this point are used to determine the values of the trigonometric functions. The cosine of an angle is defined as the ratio of the x-coordinate of the point where the terminal side intersects the unit circle to the radius of the unit circle.

Finding Cosine Using the Unit Circle

To find the cosine of an angle given the coordinates of the point where the terminal side intersects the unit circle, we can use the following formula:

cos⁡(θ)=xr\cos (\theta) = \frac{x}{r}

where xx is the x-coordinate of the point where the terminal side intersects the unit circle, and rr is the radius of the unit circle.

Example: Finding Cosine of an Angle

Let's consider an example where the terminal side of an angle in standard position intersects the unit circle at the point (−178,−478)\left(-\frac{\sqrt{17}}{8},-\frac{\sqrt{47}}{8}\right). We want to find the cosine of this angle.

Step 1: Identify the x-coordinate

The x-coordinate of the point where the terminal side intersects the unit circle is −178-\frac{\sqrt{17}}{8}.

Step 2: Identify the radius

The radius of the unit circle is 1.

Step 3: Apply the formula

Using the formula cos⁡(θ)=xr\cos (\theta) = \frac{x}{r}, we can plug in the values of xx and rr to find the cosine of the angle:

cos⁡(θ)=−1781=−178\cos (\theta) = \frac{-\frac{\sqrt{17}}{8}}{1} = -\frac{\sqrt{17}}{8}

Conclusion

In this article, we explored how to find the cosine of an angle given the coordinates of the point where the terminal side intersects the unit circle. We used the formula cos⁡(θ)=xr\cos (\theta) = \frac{x}{r} to find the cosine of an angle in standard position. We also provided an example of how to apply this formula to find the cosine of an angle.

The Importance of Understanding Trigonometric Functions

Understanding trigonometric functions is crucial in mathematics and science. Trigonometric functions are used to describe the relationships between the sides and angles of triangles, and they have numerous applications in fields such as physics, engineering, and computer science. In this article, we focused on the cosine function, but there are other trigonometric functions, such as sine and tangent, that are also important to understand.

Real-World Applications of Trigonometric Functions

Trigonometric functions have numerous real-world applications. For example, they are used in navigation systems, such as GPS, to determine the location and direction of a vehicle. They are also used in physics to describe the motion of objects, such as the trajectory of a projectile. In addition, trigonometric functions are used in computer graphics to create 3D models and animations.

Conclusion

In conclusion, the terminal side of an angle in standard position intersects the unit circle at a point, and the coordinates of this point are used to determine the values of the trigonometric functions. We used the formula cos⁡(θ)=xr\cos (\theta) = \frac{x}{r} to find the cosine of an angle in standard position. Understanding trigonometric functions is crucial in mathematics and science, and they have numerous real-world applications.

References

  • "Trigonometry" by Michael Corral
  • "Calculus" by Michael Spivak
  • "Introduction to Trigonometry" by I. M. Gelfand

Further Reading

  • "Trigonometry for Dummies" by Mary Jane Sterling
  • "Trigonometry: A Unit Circle Approach" by Charles P. McKeague
  • "Trigonometry: A Modern Approach" by L. E. Ward

Glossary

  • Unit circle: A circle with a radius of 1 centered at the origin of the coordinate plane.
  • Trigonometric functions: Functions that describe the relationships between the sides and angles of triangles.
  • Cosine: A trigonometric function that is defined as the ratio of the x-coordinate of the point where the terminal side intersects the unit circle to the radius of the unit circle.
  • Standard position: An angle is said to be in standard position if its terminal side intersects the unit circle at a point.
    The Terminal Side of an Angle in Standard Position: Q&A ===========================================================

Introduction

In our previous article, we explored how to find the cosine of an angle given the coordinates of the point where the terminal side intersects the unit circle. In this article, we will answer some frequently asked questions about the terminal side of an angle in standard position and trigonometric functions.

Q: What is the terminal side of an angle in standard position?

A: The terminal side of an angle in standard position is the line segment that extends from the origin to the point where the angle intersects 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.

Q: How do I find the cosine of an angle given the coordinates of the point where the terminal side intersects the unit circle?

A: To find the cosine of an angle given the coordinates of the point where the terminal side intersects the unit circle, you can use the formula cos⁡(θ)=xr\cos (\theta) = \frac{x}{r}, where xx is the x-coordinate of the point and rr is the radius of the unit circle.

Q: What is the significance of the unit circle in trigonometry?

A: The unit circle is a fundamental concept in trigonometry because it allows us to define the trigonometric functions of an angle. The coordinates of the point where the terminal side intersects the unit circle are used to determine the values of the trigonometric functions.

Q: Can I use the unit circle to find the sine and tangent of an angle?

A: Yes, you can use the unit circle to find the sine and tangent of an angle. The sine of an angle is defined as the ratio of the y-coordinate of the point where the terminal side intersects the unit circle to the radius of the unit circle, and the tangent of an angle is defined as the ratio of the y-coordinate of the point to the x-coordinate of the point.

Q: How do I determine the quadrant of an angle given the coordinates of the point where the terminal side intersects the unit circle?

A: To determine the quadrant of an angle given the coordinates of the point where the terminal side intersects the unit circle, you can use the following rules:

  • If the x-coordinate is positive and the y-coordinate is positive, the angle is in the first quadrant.
  • If the x-coordinate is negative and the y-coordinate is positive, the angle is in the second quadrant.
  • If the x-coordinate is negative and the y-coordinate is negative, the angle is in the third quadrant.
  • If the x-coordinate is positive and the y-coordinate is negative, the angle is in the fourth quadrant.

Q: Can I use the unit circle to find the values of the trigonometric functions of an angle in degrees?

A: Yes, you can use the unit circle to find the values of the trigonometric functions of an angle in degrees. However, you will need to convert the angle from degrees to radians first.

Q: What are some real-world applications of trigonometric functions?

A: Trigonometric functions have numerous real-world applications, including navigation systems, physics, engineering, and computer science. They are used to describe the relationships between the sides and angles of triangles, and they have numerous applications in fields such as physics, engineering, and computer science.

Conclusion

In this article, we answered some frequently asked questions about the terminal side of an angle in standard position and trigonometric functions. We hope that this article has been helpful in clarifying some of the concepts and ideas related to trigonometry.

References

  • "Trigonometry" by Michael Corral
  • "Calculus" by Michael Spivak
  • "Introduction to Trigonometry" by I. M. Gelfand

Further Reading

  • "Trigonometry for Dummies" by Mary Jane Sterling
  • "Trigonometry: A Unit Circle Approach" by Charles P. McKeague
  • "Trigonometry: A Modern Approach" by L. E. Ward

Glossary

  • Unit circle: A circle with a radius of 1 centered at the origin of the coordinate plane.
  • Trigonometric functions: Functions that describe the relationships between the sides and angles of triangles.
  • Cosine: A trigonometric function that is defined as the ratio of the x-coordinate of the point where the terminal side intersects the unit circle to the radius of the unit circle.
  • Standard position: An angle is said to be in standard position if its terminal side intersects the unit circle at a point.