Use Your Calculator To Evaluate The Following Expression: Cos ⁡ A ^ Cos ⁡ B ^ + Sin ⁡ A ^ Sin ⁡ B ^ \cos \hat{A} \cos \hat{B} + \sin \hat{A} \sin \hat{B} Cos A ^ Cos B ^ + Sin A ^ Sin B ^ If A ^ = 225 ∘ \hat{A} = 225^\circ A ^ = 22 5 ∘ And B ^ = 135 ∘ \hat{B} = 135^\circ B ^ = 13 5 ∘ .

Mar 9, 2025 by ADMIN 287 views

**Evaluating Trigonometric Expressions Using Calculators**

Introduction

Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles. It is a fundamental subject in mathematics and has numerous applications in various fields, including physics, engineering, and navigation. In this article, we will discuss how to evaluate a trigonometric expression using a calculator.

The Expression to be Evaluated

The expression to be evaluated is $\cos \hat{A} \cos \hat{B} + \sin \hat{A} \sin \hat{B}$ , where $\hat{A} = 225^\circ$ and $\hat{B} = 135^\circ$ . This expression involves the cosine and sine functions, which are fundamental trigonometric functions.

Understanding the Trigonometric Functions

Before we proceed to evaluate the expression, let's briefly review the trigonometric functions involved.

Cosine Function: The cosine function, denoted by $\cos \theta$ , is defined as the ratio of the length of the adjacent side to the length of the hypotenuse in a right-angled triangle. It is a periodic function with a period of $360^\circ$ .
Sine Function: The sine function, denoted by $\sin \theta$ , is defined as the ratio of the length of the opposite side to the length of the hypotenuse in a right-angled triangle. It is also a periodic function with a period of $360^\circ$ .

Evaluating the Expression

To evaluate the expression $\cos \hat{A} \cos \hat{B} + \sin \hat{A} \sin \hat{B}$ , we can use the following steps:

Enter the values of $\hat{A}$ and $\hat{B}$ : Enter the values of $\hat{A} = 225^\circ$ and $\hat{B} = 135^\circ$ into the calculator.
Evaluate the cosine and sine functions: Evaluate the cosine and sine functions for the given values of $\hat{A}$ and $\hat{B}$ using the calculator.
Multiply the cosine and sine functions: Multiply the cosine and sine functions for the given values of $\hat{A}$ and $\hat{B}$ .
Add the products: Add the products of the cosine and sine functions for the given values of $\hat{A}$ and $\hat{B}$ .

Using a Calculator to Evaluate the Expression

To evaluate the expression using a calculator, follow these steps:

Enter the values of $\hat{A}$ and $\hat{B}$ : Enter the values of $\hat{A} = 225^\circ$ and $\hat{B} = 135^\circ$ into the calculator.
Evaluate the cosine and sine functions: Evaluate the cosine and sine functions for the given values of $\hat{A}$ and $\hat{B}$ using the calculator.
Multiply the cosine and sine functions: Multiply the cosine and sine functions for the given values of $\hat{A}$ and $\hat{B}$ .
Add the products: Add the products of the cosine and sine functions for the given values of $\hat{A}$ and $\hat{B}$ .

Example Calculation

Let's perform an example calculation to evaluate the expression.

Enter the values of $\hat{A}$ and $\hat{B}$ : Enter the values of $\hat{A} = 225^\circ$ and $\hat{B} = 135^\circ$ into the calculator.
Evaluate the cosine and sine functions: Evaluate the cosine and sine functions for the given values of $\hat{A}$ and $\hat{B}$ using the calculator.
Multiply the cosine and sine functions: Multiply the cosine and sine functions for the given values of $\hat{A}$ and $\hat{B}$ .
Add the products: Add the products of the cosine and sine functions for the given values of $\hat{A}$ and $\hat{B}$ .

Using the Calculator to Find the Final Answer

Using the calculator, we can find the final answer to the expression.

Enter the values of $\hat{A}$ and $\hat{B}$ : Enter the values of $\hat{A} = 225^\circ$ and $\hat{B} = 135^\circ$ into the calculator.
Evaluate the cosine and sine functions: Evaluate the cosine and sine functions for the given values of $\hat{A}$ and $\hat{B}$ using the calculator.
Multiply the cosine and sine functions: Multiply the cosine and sine functions for the given values of $\hat{A}$ and $\hat{B}$ .
Add the products: Add the products of the cosine and sine functions for the given values of $\hat{A}$ and $\hat{B}$ .

The Final Answer

The final answer to the expression is $\boxed{-0.5}$ .

Conclusion

Q: What is the formula for evaluating the expression $\cos \hat{A} \cos \hat{B} + \sin \hat{A} \sin \hat{B}$ ?

A: The formula for evaluating the expression $\cos \hat{A} \cos \hat{B} + \sin \hat{A} \sin \hat{B}$ is the cosine of the difference between the two angles, i.e., $\cos (\hat{A} - \hat{B})$ .

Q: How do I enter the values of $\hat{A}$ and $\hat{B}$ into the calculator?

A: To enter the values of $\hat{A}$ and $\hat{B}$ into the calculator, simply type in the values and press the "Enter" or "Calculate" button.

Q: What if I make a mistake in entering the values of $\hat{A}$ and $\hat{B}$ ?

A: If you make a mistake in entering the values of $\hat{A}$ and $\hat{B}$ , you can simply delete the incorrect values and re-enter the correct values.

Q: Can I use a calculator to evaluate the expression for any values of $\hat{A}$ and $\hat{B}$ ?

A: Yes, you can use a calculator to evaluate the expression for any values of $\hat{A}$ and $\hat{B}$ . However, make sure to enter the values correctly and follow the steps outlined in the article.

Q: What if I get a different answer than the one provided in the article?

A: If you get a different answer than the one provided in the article, double-check your calculations and ensure that you have entered the values correctly. If you are still unsure, try re-evaluating the expression using a different calculator or method.

Q: Can I use a calculator to evaluate the expression for angles greater than $360^\circ$ ?

A: Yes, you can use a calculator to evaluate the expression for angles greater than $360^\circ$ . However, make sure to enter the values correctly and follow the steps outlined in the article.

Q: What if I need to evaluate the expression for a large number of values of $\hat{A}$ and $\hat{B}$ ?

A: If you need to evaluate the expression for a large number of values of $\hat{A}$ and $\hat{B}$ , you can use a calculator or a computer program to perform the calculations. Alternatively, you can use a trigonometric identity or formula to simplify the expression and make it easier to evaluate.

Q: Can I use a calculator to evaluate the expression for complex values of $\hat{A}$ and $\hat{B}$ ?

A: Yes, you can use a calculator to evaluate the expression for complex values of $\hat{A}$ and $\hat{B}$ . However, make sure to enter the values correctly and follow the steps outlined in the article.

Q: What if I need to evaluate the expression for a specific range of values of $\hat{A}$ and $\hat{B}$ ?

A: If you need to evaluate the expression for a specific range of values of $\hat{A}$ and $\hat{B}$ , you can use a calculator or a computer program to perform the calculations. Alternatively, you can use a trigonometric identity or formula to simplify the expression and make it easier to evaluate.

Q: Can I use a calculator to evaluate the expression for angles in radians?

A: Yes, you can use a calculator to evaluate the expression for angles in radians. However, make sure to enter the values correctly and follow the steps outlined in the article.

Q: What if I need to evaluate the expression for a large number of angles?

A: If you need to evaluate the expression for a large number of angles, you can use a calculator or a computer program to perform the calculations. Alternatively, you can use a trigonometric identity or formula to simplify the expression and make it easier to evaluate.

Q: Can I use a calculator to evaluate the expression for angles in degrees and radians?

A: Yes, you can use a calculator to evaluate the expression for angles in degrees and radians. However, make sure to enter the values correctly and follow the steps outlined in the article.

Q: What if I need to evaluate the expression for a specific range of angles?

A: If you need to evaluate the expression for a specific range of angles, you can use a calculator or a computer program to perform the calculations. Alternatively, you can use a trigonometric identity or formula to simplify the expression and make it easier to evaluate.