Given The Ordered Pair \[$P(6, -5)\$\], Determine The Requested \[$\sin\$\], \[$\cos\$\], Or \[$\tan\$\] Of Angle \[$\theta\$\].\[$\tan \theta =\$\]A. \[$ \frac{6}{5} \$\]B. \[$ -\frac{6}{5}

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Introduction

Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles. It is a fundamental subject that has numerous applications in various fields, including physics, engineering, and navigation. In this article, we will explore how to determine the requested trigonometric ratios, such as sine, cosine, and tangent, from an ordered pair.

What are Trigonometric Ratios?

Trigonometric ratios are the ratios of the lengths of the sides of a right triangle to the lengths of the sides adjacent to the angle. The three basic trigonometric ratios are:

  • Sine (sin): The ratio of the length of the side opposite the angle to the length of the hypotenuse.
  • Cosine (cos): The ratio of the length of the side adjacent to the angle to the length of the hypotenuse.
  • Tangent (tan): The ratio of the length of the side opposite the angle to the length of the side adjacent to the angle.

Ordered Pairs and Trigonometric Ratios

An ordered pair is a pair of numbers that are used to represent a point on a coordinate plane. In the context of trigonometry, an ordered pair can be used to represent the coordinates of a point on the terminal side of an angle. The ordered pair is typically written in the form (x, y), where x is the x-coordinate and y is the y-coordinate.

Determining Trigonometric Ratios from Ordered Pairs

To determine the requested trigonometric ratio from an ordered pair, we need to use the following formulas:

  • Sine (sin): sin(θ) = y / r, where y is the y-coordinate and r is the distance from the origin to the point (x, y).
  • Cosine (cos): cos(θ) = x / r, where x is the x-coordinate and r is the distance from the origin to the point (x, y).
  • Tangent (tan): tan(θ) = y / x, where y is the y-coordinate and x is the x-coordinate.

Example: Determining the Tangent of an Angle from an Ordered Pair

Given the ordered pair P(6, -5), we can use the formula tan(θ) = y / x to determine the tangent of the angle θ.

Step 1: Identify the x and y coordinates

The x-coordinate is 6 and the y-coordinate is -5.

Step 2: Plug in the values into the formula

tan(θ) = -5 / 6

Step 3: Simplify the expression

tan(θ) = -5/6

Therefore, the tangent of the angle θ is -5/6.

Conclusion

In this article, we have explored how to determine the requested trigonometric ratios, such as sine, cosine, and tangent, from an ordered pair. We have used the formulas sin(θ) = y / r, cos(θ) = x / r, and tan(θ) = y / x to determine the trigonometric ratios from the ordered pair P(6, -5). We have also provided an example of how to determine the tangent of an angle from an ordered pair.

References

  • "Trigonometry" by Michael Corral
  • "Precalculus" by James Stewart

Further Reading

  • "Trigonometry for Dummies" by Mary Jane Sterling
  • "Precalculus: Mathematics for Calculus" by James Stewart

Glossary

  • Ordered pair: A pair of numbers that are used to represent a point on a coordinate plane.
  • Trigonometric ratio: The ratio of the lengths of the sides of a right triangle to the lengths of the sides adjacent to the angle.
  • Sine (sin): The ratio of the length of the side opposite the angle to the length of the hypotenuse.
  • Cosine (cos): The ratio of the length of the side adjacent to the angle to the length of the hypotenuse.
  • Tangent (tan): The ratio of the length of the side opposite the angle to the length of the side adjacent to the angle.
    Trigonometric Ratios from Ordered Pairs: Q&A =============================================

Introduction

In our previous article, we explored how to determine the requested trigonometric ratios, such as sine, cosine, and tangent, from an ordered pair. In this article, we will answer some frequently asked questions about trigonometric ratios from ordered pairs.

Q: What is the difference between an ordered pair and a coordinate point?

A: An ordered pair is a pair of numbers that are used to represent a point on a coordinate plane. A coordinate point is a point on a coordinate plane that is represented by an ordered pair.

Q: How do I determine the trigonometric ratios from an ordered pair?

A: To determine the trigonometric ratios from an ordered pair, you need to use the following formulas:

  • Sine (sin): sin(θ) = y / r, where y is the y-coordinate and r is the distance from the origin to the point (x, y).
  • Cosine (cos): cos(θ) = x / r, where x is the x-coordinate and r is the distance from the origin to the point (x, y).
  • Tangent (tan): tan(θ) = y / x, where y is the y-coordinate and x is the x-coordinate.

Q: What is the distance from the origin to a point on a coordinate plane?

A: The distance from the origin to a point on a coordinate plane is called the radius or the distance from the origin. It can be calculated using the formula r = √(x^2 + y^2), where x is the x-coordinate and y is the y-coordinate.

Q: Can I use the trigonometric ratios to solve problems involving right triangles?

A: Yes, you can use the trigonometric ratios to solve problems involving right triangles. The trigonometric ratios can be used to find the lengths of the sides of a right triangle, as well as the measures of the angles.

Q: What are some common mistakes to avoid when working with trigonometric ratios?

A: Some common mistakes to avoid when working with trigonometric ratios include:

  • Not using the correct formula: Make sure to use the correct formula for the trigonometric ratio you are trying to find.
  • Not simplifying the expression: Make sure to simplify the expression before giving the final answer.
  • Not checking the units: Make sure to check the units of the answer to ensure that they are correct.

Q: Can I use trigonometric ratios to solve problems involving non-right triangles?

A: Yes, you can use trigonometric ratios to solve problems involving non-right triangles. However, you will need to use the Law of Sines or the Law of Cosines to find the lengths of the sides of the triangle.

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

A: Some real-world applications of trigonometric ratios include:

  • Navigation: Trigonometric ratios are used in navigation to find the distance and direction between two points.
  • Physics: Trigonometric ratios are used in physics to describe the motion of objects.
  • Engineering: Trigonometric ratios are used in engineering to design and build structures.

Conclusion

In this article, we have answered some frequently asked questions about trigonometric ratios from ordered pairs. We have also discussed some common mistakes to avoid when working with trigonometric ratios, as well as some real-world applications of trigonometric ratios.

References

  • "Trigonometry" by Michael Corral
  • "Precalculus" by James Stewart

Further Reading

  • "Trigonometry for Dummies" by Mary Jane Sterling
  • "Precalculus: Mathematics for Calculus" by James Stewart

Glossary

  • Ordered pair: A pair of numbers that are used to represent a point on a coordinate plane.
  • Trigonometric ratio: The ratio of the lengths of the sides of a right triangle to the lengths of the sides adjacent to the angle.
  • Sine (sin): The ratio of the length of the side opposite the angle to the length of the hypotenuse.
  • Cosine (cos): The ratio of the length of the side adjacent to the angle to the length of the hypotenuse.
  • Tangent (tan): The ratio of the length of the side opposite the angle to the length of the side adjacent to the angle.