$\csc \left(\arctan \left(\frac{9}{7}\right)\right) =$
Introduction
Trigonometric equations are a fundamental aspect of mathematics, and solving them requires a deep understanding of various trigonometric functions and their relationships. In this article, we will focus on evaluating the expression , which involves a combination of inverse trigonometric functions and the cosecant function. We will break down the solution into manageable steps, using a combination of mathematical reasoning and trigonometric identities to arrive at the final answer.
Understanding the Components
Before we dive into the solution, let's take a closer look at the components involved in the expression . The expression consists of two main components:
- Arctan: The arctan function, also known as the inverse tangent function, is a function that returns the angle whose tangent is a given number. In this case, we have , which means we are looking for the angle whose tangent is .
- Cosecant: The cosecant function, denoted by , is the reciprocal of the sine function. It is defined as .
Step 1: Evaluating the Arctan Function
To evaluate the expression , we need to start by evaluating the arctan function. The arctan function returns the angle whose tangent is a given number. In this case, we have , which means we are looking for the angle whose tangent is .
Using a calculator or a trigonometric table, we can find that the angle whose tangent is is approximately radians. However, we can also express this angle in terms of a known trigonometric identity.
Step 2: Using Trigonometric Identities
We can use the trigonometric identity to rewrite the expression in terms of sine and cosine.
Let . Then, we can write:
Step 3: Finding the Sine and Cosine Values
Using the trigonometric identity , we can rewrite the expression as a ratio of sine and cosine values.
Let and , where is a constant. Then, we can write:
Step 4: Using the Pythagorean Identity
We can use the Pythagorean identity to find the value of .
Substituting the expressions for and , we get:
Simplifying the equation, we get:
Combine like terms:
Divide both sides by 130:
Take the square root of both sides:
Step 5: Finding the Sine and Cosine Values
Now that we have found the value of , we can substitute it back into the expressions for and .
Step 6: Evaluating the Cosecant Function
Now that we have found the values of and , we can evaluate the cosecant function.
Simplify the expression:
Rationalize the denominator:
Simplify the expression:
Conclusion
In this article, we have evaluated the expression using a combination of trigonometric identities and mathematical reasoning. We have broken down the solution into manageable steps, starting with the evaluation of the arctan function and ending with the evaluation of the cosecant function.
The final answer is .
Q: What is the arctan function?
A: The arctan function, also known as the inverse tangent function, is a function that returns the angle whose tangent is a given number. In this case, we have , which means we are looking for the angle whose tangent is .
Q: How do I evaluate the arctan function?
A: To evaluate the arctan function, you can use a calculator or a trigonometric table. Alternatively, you can use the trigonometric identity to rewrite the expression in terms of sine and cosine.
Q: What is the cosecant function?
A: The cosecant function, denoted by , is the reciprocal of the sine function. It is defined as .
Q: How do I evaluate the cosecant function?
A: To evaluate the cosecant function, you need to find the value of and then take its reciprocal. In this case, we have , so we can evaluate the cosecant function as .
Q: What is the final answer to the expression ?
A: The final answer to the expression is .
Q: Can I use a calculator to evaluate the expression ?
A: Yes, you can use a calculator to evaluate the expression . However, it's also important to understand the underlying mathematical concepts and trigonometric identities that are used to evaluate the expression.
Q: What is the significance of the Pythagorean identity in evaluating the expression ?
A: The Pythagorean identity is used to find the value of in the expression and . This identity is a fundamental concept in trigonometry and is used to relate the sine and cosine functions.
Q: Can I use the expression in real-world applications?
A: Yes, the expression can be used in real-world applications such as engineering, physics, and computer science. For example, it can be used to model the behavior of waves, vibrations, and other oscillatory phenomena.
Q: How do I simplify the expression ?
A: To simplify the expression , you can use a combination of trigonometric identities and mathematical reasoning. Start by evaluating the arctan function, then use the Pythagorean identity to find the value of , and finally evaluate the cosecant function.