Rewrite \csc \left(\cos ^{-1} 3v\right ] As An Algebraic Expression In V V V .

Introduction

Trigonometric functions are a fundamental part of mathematics, and they play a crucial role in various fields such as physics, engineering, and computer science. In this article, we will focus on rewriting a specific trigonometric expression involving the cosecant function and the inverse cosine function. We will break down the problem into manageable steps and provide a clear explanation of each step.

The Problem

The given expression is $\csc \left(\cos ^{-1} 3v\right)$. Our goal is to rewrite this expression as an algebraic expression in terms of $v$.

Step 1: Understanding the Inverse Cosine Function

The inverse cosine function, denoted as $\cos^{-1} x$, is the angle whose cosine is equal to $x$. In other words, if $\cos \theta = x$, then $\cos^{-1} x = \theta$. This function is also known as the arccosine function.

Step 2: Analyzing the Given Expression

The given expression is $\csc \left(\cos ^{-1} 3v\right)$. We can see that the argument of the cosecant function is the inverse cosine of $3v$. Our goal is to rewrite this expression in terms of $v$.

Step 3: Using the Definition of the Cosecant Function

The cosecant function is defined as the reciprocal of the sine function, i.e., $\csc \theta = \frac{1}{\sin \theta}$. We can use this definition to rewrite the given expression.

Step 4: Rewriting the Expression

Using the definition of the cosecant function, we can rewrite the given expression as follows:

$$\csc \left(\cos ^{-1} 3v\right) = \frac{1}{\sin \left(\cos ^{-1} 3v\right)}$$

Step 5: Simplifying the Expression

We can simplify the expression by using the fact that the sine and cosine functions are related by the Pythagorean identity, i.e., $\sin^2 \theta + \cos^2 \theta = 1$. We can rewrite the expression as follows:

$$\frac{1}{\sin \left(\cos ^{-1} 3v\right)} = \frac{1}{\sqrt{1 - \cos^2 \left(\cos ^{-1} 3v\right)}}$$

Step 6: Evaluating the Expression

We can evaluate the expression by using the fact that the cosine function is equal to $3v$ in the given expression. We can rewrite the expression as follows:

$$\frac{1}{\sqrt{1 - \cos^2 \left(\cos ^{-1} 3v\right)}} = \frac{1}{\sqrt{1 - (3v)^2}}$$

Step 7: Simplifying the Expression

We can simplify the expression by evaluating the square root:

$$\frac{1}{\sqrt{1 - (3v)^2}} = \frac{1}{\sqrt{1 - 9v^2}}$$

Step 8: Rationalizing the Denominator

We can rationalize the denominator by multiplying both the numerator and the denominator by the conjugate of the denominator:

$$\frac{1}{\sqrt{1 - 9v^2}} = \frac{1}{\sqrt{1 - 9v^2}} \cdot \frac{\sqrt{1 - 9v^2}}{\sqrt{1 - 9v^2}}$$

Step 9: Simplifying the Expression

We can simplify the expression by canceling out the common factors:

$$\frac{1}{\sqrt{1 - 9v^2}} \cdot \frac{\sqrt{1 - 9v^2}}{\sqrt{1 - 9v^2}} = \frac{\sqrt{1 - 9v^2}}{1 - 9v^2}$$

Step 10: Final Answer

The final answer is $\boxed{\frac{\sqrt{1 - 9v^2}}{1 - 9v^2}}$.

Conclusion

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Introduction

In our previous article, we rewrote a trigonometric expression involving the cosecant function and the inverse cosine function as an algebraic expression in terms of $v$. In this article, we will provide a Q&A section to help clarify any doubts or questions that readers may have.

Q: What is the inverse cosine function?

A: The inverse cosine function, denoted as $\cos^{-1} x$, is the angle whose cosine is equal to $x$. In other words, if $\cos \theta = x$, then $\cos^{-1} x = \theta$.

Q: How do I evaluate the expression $\csc \left(\cos ^{-1} 3v\right)$?

A: To evaluate the expression, we can use the definition of the cosecant function, which is the reciprocal of the sine function. We can rewrite the expression as $\frac{1}{\sin \left(\cos ^{-1} 3v\right)}$.

Q: What is the Pythagorean identity?

A: The Pythagorean identity is a fundamental concept in trigonometry, which states that $\sin^2 \theta + \cos^2 \theta = 1$. We can use this identity to simplify the expression.

Q: How do I simplify the expression $\frac{1}{\sqrt{1 - \cos^2 \left(\cos ^{-1} 3v\right)}}$?

A: To simplify the expression, we can evaluate the square root and then rationalize the denominator.

Q: What is rationalizing the denominator?

A: Rationalizing the denominator involves multiplying both the numerator and the denominator by the conjugate of the denominator. This helps to eliminate any radicals in the denominator.

Q: How do I rationalize the denominator of the expression $\frac{1}{\sqrt{1 - 9v^2}}$?

A: To rationalize the denominator, we can multiply both the numerator and the denominator by the conjugate of the denominator, which is $\sqrt{1 - 9v^2}$.

Q: What is the final answer to the expression $\csc \left(\cos ^{-1} 3v\right)$?

A: The final answer to the expression is $\boxed{\frac{\sqrt{1 - 9v^2}}{1 - 9v^2}}$.

Q: Can I use this method to rewrite other trigonometric expressions?

A: Yes, the method we used to rewrite the expression $\csc \left(\cos ^{-1} 3v\right)$ can be applied to other trigonometric expressions involving the cosecant function and the inverse cosine function.

Q: Are there any other trigonometric identities that I should know?

A: Yes, there are many other trigonometric identities that you should know, such as the sine and cosine addition formulas, the double-angle formulas, and the half-angle formulas.

Conclusion

In this Q&A article, we have provided answers to common questions and doubts that readers may have about rewriting trigonometric expressions involving the cosecant function and the inverse cosine function. We hope that this article has been helpful in clarifying any confusion and providing a better understanding of the subject.