Evaluating the Cosine Function

Oct 13, 2025 by windailys.com 115 views

Introduction

In this article, we will delve into the world of trigonometry and explore a problem that involves evaluating and simplifying a trigonometric expression. The problem requires us to find the value of $c$ in the given equation, which involves the cosine function and some basic algebraic manipulations. We will break down the problem step by step, using the properties of trigonometric functions and algebraic techniques to simplify the expression and arrive at the final answer.

Understanding the Problem

The problem presents us with an equation that involves the cosine function and some basic algebraic manipulations. The equation is given as:

c^2 = 3^2 + 10^2 - 2(3)(10) \cos \left(34^{\circ}\right)

Our goal is to find the value of $c$ by evaluating and simplifying this expression. To do this, we need to understand the properties of the cosine function and how it behaves in different ranges of angles.

Evaluating the Cosine Function

The cosine function is a periodic function that oscillates between -1 and 1. It is defined as the ratio of the adjacent side to the hypotenuse in a right-angled triangle. In this problem, we are given an angle of $34^{\circ}$ , and we need to evaluate the cosine of this angle.

Using a calculator or a trigonometric table, we can find that the cosine of $34^{\circ}$ is approximately 0.8290.

Substituting the Value of Cosine

Now that we have evaluated the cosine function, we can substitute this value into the original equation:

c^2 = 3^2 + 10^2 - 2(3)(10) \cos \left(34^{\circ}\right)

Substituting the value of cosine, we get:

c^2 = 3^2 + 10^2 - 2(3)(10)(0.8290)

Simplifying the Expression

Now that we have substituted the value of cosine, we can simplify the expression by performing the arithmetic operations.

First, we can calculate the squares of 3 and 10:

3^2 = 9

10^2 = 100

Next, we can calculate the product of 2, 3, and 10:

2(3)(10) = 60

Now, we can substitute these values into the expression:

c^2 = 9 + 100 - 60(0.8290)

Evaluating the Expression

Now that we have simplified the expression, we can evaluate it by performing the arithmetic operations.

First, we can calculate the product of 60 and 0.8290:

60(0.8290) = 49.74

Next, we can substitute this value into the expression:

c^2 = 9 + 100 - 49.74

Finding the Value of $c$

Now that we have evaluated the expression, we can find the value of $c$ by taking the square root of both sides of the equation.

c^2 = 59.26

Taking the square root of both sides, we get:

c = \sqrt{59.26}

Rounding to the Nearest Tenth

Finally, we need to round the value of $c$ to the nearest tenth.

c = 7.7

Therefore, the value of $c$ is approximately 7.7.

Conclusion

In this article, we have evaluated and simplified a trigonometric expression involving the cosine function. We have used the properties of trigonometric functions and algebraic techniques to simplify the expression and arrive at the final answer. We have also rounded the value of $c$ to the nearest tenth, as required by the problem. The final answer is 7.7.

Additional Resources

For more information on trigonometry and algebra, please refer to the following resources:

Introduction

In our previous article, we explored a problem that involved evaluating and simplifying a trigonometric expression. We broke down the problem step by step, using the properties of trigonometric functions and algebraic techniques to simplify the expression and arrive at the final answer. In this article, we will answer some common questions that readers may have about the problem.

Q&A

Q: What is the significance of the cosine function in this problem?

A: The cosine function is used to evaluate the expression involving the angle of $34^{\circ}$ . The cosine function is a periodic function that oscillates between -1 and 1, and it is defined as the ratio of the adjacent side to the hypotenuse in a right-angled triangle.

Q: How do I evaluate the cosine function?

A: You can use a calculator or a trigonometric table to evaluate the cosine function. Alternatively, you can use the unit circle or the trigonometric identities to evaluate the cosine function.

Q: What is the difference between the cosine function and the sine function?

A: The cosine function and the sine function are both trigonometric functions that are used to evaluate the ratio of the sides of a right-angled triangle. However, the cosine function is defined as the ratio of the adjacent side to the hypotenuse, while the sine function is defined as the ratio of the opposite side to the hypotenuse.

Q: How do I simplify the expression involving the cosine function?

A: To simplify the expression involving the cosine function, you can use the properties of trigonometric functions and algebraic techniques. You can also use the trigonometric identities to simplify the expression.

Q: What is the final answer to the problem?

A: The final answer to the problem is $c = 7.7$ . This is the value of $c$ that we obtained by evaluating and simplifying the expression involving the cosine function.

Q: How do I round the value of $c$ to the nearest tenth?

A: To round the value of $c$ to the nearest tenth, you can use the rounding rules. If the digit in the hundredths place is 5 or greater, you round up. If the digit in the hundredths place is less than 5, you round down.

Q: What are some common mistakes that people make when evaluating and simplifying trigonometric expressions?

A: Some common mistakes that people make when evaluating and simplifying trigonometric expressions include:

Not using the correct trigonometric identities
Not simplifying the expression correctly
Not using the correct rounding rules
Not checking the units of the answer

Conclusion

In this article, we have answered some common questions that readers may have about the problem involving evaluating and simplifying a trigonometric expression. We have also provided some tips and tricks for evaluating and simplifying trigonometric expressions. By following these tips and tricks, you can avoid common mistakes and arrive at the correct answer.

Additional Resources

For more information on trigonometry and algebra, please refer to the following resources:

Frequently Asked Questions

Q: What is the difference between the cosine function and the sine function? A: The cosine function and the sine function are both trigonometric functions that are used to evaluate the ratio of the sides of a right-angled triangle. However, the cosine function is defined as the ratio of the adjacent side to the hypotenuse, while the sine function is defined as the ratio of the opposite side to the hypotenuse.
Q: How do I simplify the expression involving the cosine function? A: To simplify the expression involving the cosine function, you can use the properties of trigonometric functions and algebraic techniques. You can also use the trigonometric identities to simplify the expression.
Q: What is the final answer to the problem? A: The final answer to the problem is $c = 7.7$ . This is the value of $c$ that we obtained by evaluating and simplifying the expression involving the cosine function.
Q: How do I round the value of $c$ to the nearest tenth? A: To round the value of $c$ to the nearest tenth, you can use the rounding rules. If the digit in the hundredths place is 5 or greater, you round up. If the digit in the hundredths place is less than 5, you round down.