Let $(-3, \sqrt{7})$ Be A Point On The Terminal Side Of $\theta$. Find The Exact Values Of $\sin \theta$, $\csc \theta$, And $\cot \theta$.

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 focus on finding the exact values of trigonometric functions, specifically $\sin \theta$, $\csc \theta$, and $\cot \theta$, given a point on the terminal side of $\theta$.

The Problem

Let $(-3, \sqrt{7})$ be a point on the terminal side of $\theta$. Our goal is to find the exact values of $\sin \theta$, $\csc \theta$, and $\cot \theta$.

Recall: Trigonometric Functions

Before we dive into the problem, let's recall the definitions of the trigonometric functions:

• $\sin \theta = \frac{y}{r}$, where $y$ is the $y$-coordinate of the point on the terminal side of $\theta$ and $r$ is the distance from the origin to the point.
• $\csc \theta = \frac{1}{\sin \theta} = \frac{r}{y}$, where $y$ is the $y$-coordinate of the point on the terminal side of $\theta$ and $r$ is the distance from the origin to the point.
• $\cot \theta = \frac{\cos \theta}{\sin \theta} = \frac{x}{y}$, where $x$ is the $x$-coordinate of the point on the terminal side of $\theta$ and $y$ is the $y$-coordinate of the point.

Finding the Distance from the Origin

To find the exact values of $\sin \theta$, $\csc \theta$, and $\cot \theta$, we need to find the distance from the origin to the point $(-3, \sqrt{7})$. We can use the distance formula:

$r = \sqrt{x^2 + y^2}$

where $x$ and $y$ are the coordinates of the point.

Plugging in the values, we get:

$r = \sqrt{(-3)^2 + (\sqrt{7})^2}$ $r = \sqrt{9 + 7}$ $r = \sqrt{16}$ $r = 4$

Finding the Exact Values of Trigonometric Functions

Now that we have the distance from the origin, we can find the exact values of $\sin \theta$, $\csc \theta$, and $\cot \theta$.

• $\sin \theta = \frac{y}{r} = \frac{\sqrt{7}}{4}$
• $\csc \theta = \frac{1}{\sin \theta} = \frac{4}{\sqrt{7}} = \frac{4\sqrt{7}}{7}$
• $\cot \theta = \frac{x}{y} = \frac{-3}{\sqrt{7}} = \frac{-3\sqrt{7}}{7}$

Conclusion

In this article, we found the exact values of $\sin \theta$, $\csc \theta$, and $\cot \theta$ given a point on the terminal side of $\theta$. We used the distance formula to find the distance from the origin to the point and then used the definitions of the trigonometric functions to find the exact values. We hope this article has provided a clear understanding of how to find the exact values of trigonometric functions.

References

• [1] "Trigonometry" by Michael Corral
• [2] "Precalculus" by James Stewart

Glossary

• Trigonometry: A branch of mathematics that deals with the relationships between the sides and angles of triangles.
• Terminal side: The side of a triangle that contains the angle $\theta$.
• Distance formula: A formula used to find the distance between two points in a coordinate plane.
• Trigonometric functions: Functions that relate the sides and angles of triangles, including $\sin \theta$, $\csc \theta$, and $\cot \theta$.
  Trigonometric Functions: Q&A =============================

Introduction

In our previous article, we discussed how to find the exact values of trigonometric functions, specifically $\sin \theta$, $\csc \theta$, and $\cot \theta$, given a point on the terminal side of $\theta$. In this article, we will answer some common questions related to trigonometric functions.

Q: What is the difference between $\sin \theta$ and $\csc \theta$?

A: $\sin \theta$ is the ratio of the $y$-coordinate of a point on the terminal side of $\theta$ to the distance from the origin to the point, while $\csc \theta$ is the reciprocal of $\sin \theta$. In other words, $\csc \theta = \frac{1}{\sin \theta}$.

Q: How do I find the value of $\cot \theta$?

A: To find the value of $\cot \theta$, you need to find the value of $\cos \theta$ and then divide it by the value of $\sin \theta$. In other words, $\cot \theta = \frac{\cos \theta}{\sin \theta}$.

Q: What is the relationship between $\sin \theta$ and $\cos \theta$?

A: The sine and cosine functions are related by the Pythagorean identity: $\sin^2 \theta + \cos^2 \theta = 1$. This means that if you know the value of one of these functions, you can find the value of the other.

Q: Can I use a calculator to find the values of trigonometric functions?

A: Yes, you can use a calculator to find the values of trigonometric functions. However, keep in mind that calculators may not always give you the exact values, especially for irrational numbers.

Q: How do I find the value of $\tan \theta$?

A: To find the value of $\tan \theta$, you need to find the value of $\sin \theta$ and then divide it by the value of $\cos \theta$. In other words, $\tan \theta = \frac{\sin \theta}{\cos \theta}$.

Q: What is the relationship between $\sin \theta$ and $\csc \theta$?

A: The sine and cosecant functions are reciprocals of each other. In other words, $\csc \theta = \frac{1}{\sin \theta}$.

Q: Can I use trigonometric functions to solve real-world problems?

A: Yes, trigonometric functions have numerous applications in real-world problems, such as navigation, physics, engineering, and more.

Q: How do I choose the correct trigonometric function to use in a problem?

A: To choose the correct trigonometric function, you need to identify the relationship between the sides and angles of the triangle. For example, if you are given the length of the hypotenuse and the length of one of the legs, you would use the sine function.

Conclusion

In this article, we answered some common questions related to trigonometric functions. We hope this article has provided a clear understanding of how to use trigonometric functions to solve problems.

References

• [1] "Trigonometry" by Michael Corral
• [2] "Precalculus" by James Stewart

Glossary

• Trigonometric functions: Functions that relate the sides and angles of triangles, including $\sin \theta$, $\csc \theta$, and $\cot \theta$.
• Terminal side: The side of a triangle that contains the angle $\theta$.
• Distance formula: A formula used to find the distance between two points in a coordinate plane.
• Pythagorean identity: A mathematical identity that relates the sine and cosine functions: $\sin^2 \theta + \cos^2 \theta = 1$.

Additional Resources

• [1] Khan Academy: Trigonometry
• [2] MIT OpenCourseWare: Trigonometry
• [3] Wolfram Alpha: Trigonometry