Verify The Identity Algebraically. Use A Graphing Utility To Check Your Result Graphically. 4 Sec ⁡ ( Y ) Cos ⁡ ( Y ) = 4 4 \sec(y) \cos(y) = 4 4 Sec ( Y ) Cos ( Y ) = 4 4 Sec ⁡ ( Y ) Cos ⁡ ( Y ) = ( 4 ) Cos ⁡ ( Y ) = 4 4 \sec(y) \cos(y) = \left(\frac{4}{\sqrt{~}}\right) \cos(y) = 4 4 Sec ( Y ) Cos ( Y ) = ( ​ 4 ​ ) Cos ( Y ) = 4

Introduction

In this article, we will explore the process of verifying an identity algebraically and graphically. We will focus on the given trigonometric identity: $4 \sec(y) \cos(y) = 4$. Our goal is to simplify the left-hand side of the equation and show that it is equivalent to the right-hand side. We will also use a graphing utility to check our result graphically.

Algebraic Verification

To verify the identity algebraically, we need to simplify the left-hand side of the equation. We can start by rewriting the secant function in terms of the cosine function:

$$\sec(y) = \frac{1}{\cos(y)}$$

Substituting this expression into the original equation, we get:

$$4 \sec(y) \cos(y) = 4 \left(\frac{1}{\cos(y)}\right) \cos(y)$$

Simplifying the expression, we get:

$$4 \sec(y) \cos(y) = 4$$

This shows that the left-hand side of the equation is indeed equal to the right-hand side.

Graphical Verification

To verify the identity graphically, we can use a graphing utility to plot the functions $y = 4 \sec(y) \cos(y)$ and $y = 4$. If the two graphs are identical, then the identity is verified.

Using a graphing utility, we can plot the two functions and observe that they are indeed identical. This provides further evidence that the identity is true.

Discussion

In this article, we have shown that the identity $4 \sec(y) \cos(y) = 4$ can be verified both algebraically and graphically. The algebraic verification involved simplifying the left-hand side of the equation using trigonometric identities, while the graphical verification involved plotting the two functions using a graphing utility.

Conclusion

In conclusion, verifying an identity algebraically and graphically is an important step in mathematics. It provides a way to check the validity of an equation and ensures that it is true for all values of the variable. In this article, we have shown that the identity $4 \sec(y) \cos(y) = 4$ can be verified using both methods, and we have provided a clear and concise explanation of the process.

Step-by-Step Solution

Here is a step-by-step solution to the problem:

1. Rewrite the secant function in terms of the cosine function: $\sec(y) = \frac{1}{\cos(y)}$
2. Substitute this expression into the original equation: $4 \sec(y) \cos(y) = 4 \left(\frac{1}{\cos(y)}\right) \cos(y)$
3. Simplify the expression: $4 \sec(y) \cos(y) = 4$
4. Use a graphing utility to plot the functions $y = 4 \sec(y) \cos(y)$ and $y = 4$
5. Observe that the two graphs are identical

Tips and Variations

Here are some tips and variations to the problem:

• To verify the identity for a specific value of $y$, substitute that value into the equation and simplify.
• To verify the identity for a range of values of $y$, use a graphing utility to plot the functions and observe that they are identical.
• To verify the identity for a different trigonometric function, substitute that function into the equation and simplify.

Common Mistakes

Here are some common mistakes to avoid when verifying an identity:

• Failing to simplify the left-hand side of the equation
• Failing to use a graphing utility to check the result graphically
• Assuming that the identity is true without verifying it algebraically and graphically

Real-World Applications

Here are some real-world applications of verifying an identity:

• In physics, identities are used to simplify complex equations and make them easier to solve.
• In engineering, identities are used to design and optimize systems.
• In computer science, identities are used to write efficient and effective code.

Conclusion

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Introduction

In our previous article, we explored the process of verifying an identity algebraically and graphically. We focused on the given trigonometric identity: $4 \sec(y) \cos(y) = 4$. In this article, we will answer some common questions related to verifying identities and provide additional tips and resources.

Q: What is the purpose of verifying an identity?

A: The purpose of verifying an identity is to ensure that it is true for all values of the variable. This is an important step in mathematics, as it helps to build confidence in the validity of an equation and ensures that it can be used to solve problems.

Q: How do I verify an identity algebraically?

A: To verify an identity algebraically, you need to simplify the left-hand side of the equation using trigonometric identities. This may involve rewriting the secant function in terms of the cosine function, or using other trigonometric identities to simplify the expression.

Q: How do I verify an identity graphically?

A: To verify an identity graphically, you need to use a graphing utility to plot the functions on both sides of the equation. If the two graphs are identical, then the identity is verified.

Q: What are some common mistakes to avoid when verifying an identity?

A: Some common mistakes to avoid when verifying an identity include:

• Failing to simplify the left-hand side of the equation
• Failing to use a graphing utility to check the result graphically
• Assuming that the identity is true without verifying it algebraically and graphically

Q: How do I use a graphing utility to verify an identity?

A: To use a graphing utility to verify an identity, follow these steps:

1. Enter the functions on both sides of the equation into the graphing utility.
2. Set the window to a suitable range of values for the variable.
3. Plot the functions using the graphing utility.
4. Observe that the two graphs are identical.

Q: What are some real-world applications of verifying identities?

A: Some real-world applications of verifying identities include:

• In physics, identities are used to simplify complex equations and make them easier to solve.
• In engineering, identities are used to design and optimize systems.
• In computer science, identities are used to write efficient and effective code.

Q: How do I choose the right trigonometric identity to use when verifying an identity?

A: When choosing the right trigonometric identity to use when verifying an identity, consider the following:

• Look for identities that involve the secant and cosine functions.
• Consider using the Pythagorean identity: $\sec^2(y) - \tan^2(y) = 1$
• Consider using the identity: $\sec(y) = \frac{1}{\cos(y)}$

Q: What are some additional resources for learning about verifying identities?

A: Some additional resources for learning about verifying identities include:

• Online tutorials and videos
• Textbooks and workbooks
• Online communities and forums
• Graphing utility software and apps

Conclusion

In conclusion, verifying an identity algebraically and graphically is an important step in mathematics. It provides a way to check the validity of an equation and ensures that it is true for all values of the variable. In this article, we have answered some common questions related to verifying identities and provided additional tips and resources.