Evaluate The Expression:$ 2 \cdot \sin 45^{\circ}\left(\tan 45^{\circ}-\cos 60^{\circ}\right) }$Note The Following Identities May Be Useful:- { \sin 45^{\circ = \frac{\sqrt{2}}{2}$}$- { \tan 45^{\circ} = 1$}$-

Introduction

Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles. It is a fundamental subject that has numerous applications in various fields, including physics, engineering, and navigation. In this article, we will focus on evaluating a specific trigonometric expression using the given identities.

The Expression to Evaluate

The expression to evaluate is:

$$2 \cdot \sin 45^{\circ}\left(\tan 45^{\circ}-\cos 60^{\circ}\right)$$

We are given the following identities:

• $\sin 45^{\circ} = \frac{\sqrt{2}}{2}$
• $\tan 45^{\circ} = 1$
• $\cos 60^{\circ} = \frac{1}{2}$

Step 1: Substitute the Given Identities

We will start by substituting the given identities into the expression.

$$2 \cdot \frac{\sqrt{2}}{2}\left(1-\frac{1}{2}\right)$$

Step 2: Simplify the Expression

Next, we will simplify the expression by performing the arithmetic operations.

$$2 \cdot \frac{\sqrt{2}}{2}\left(\frac{1}{2}\right)$$

$$\frac{\sqrt{2}}{2} \cdot \frac{1}{2}$$

$$\frac{\sqrt{2}}{4}$$

Step 3: Rationalize the Denominator (Optional)

In this case, we do not need to rationalize the denominator because the expression is already simplified.

Conclusion

In this article, we evaluated the given trigonometric expression using the provided identities. We substituted the identities into the expression, simplified it, and obtained the final result. This example demonstrates the importance of using trigonometric identities to simplify complex expressions.

Common Trigonometric Identities

Here are some common trigonometric identities that may be useful in evaluating expressions:

• $\sin^2 x + \cos^2 x = 1$
• $\tan x = \frac{\sin x}{\cos x}$
• $\cot x = \frac{\cos x}{\sin x}$
• $\sec x = \frac{1}{\cos x}$
• $\csc x = \frac{1}{\sin x}$

Tips and Tricks

Here are some tips and tricks for evaluating trigonometric expressions:

• Use the given identities to simplify the expression.
• Perform the arithmetic operations in the correct order.
• Simplify the expression by combining like terms.
• Rationalize the denominator if necessary.

Practice Problems

Here are some practice problems to help you evaluate trigonometric expressions:

• Evaluate the expression: $2 \cdot \sin 30^{\circ}\left(\tan 30^{\circ}-\cos 90^{\circ}\right)$
• Evaluate the expression: $3 \cdot \cos 45^{\circ}\left(\sin 45^{\circ}-\tan 60^{\circ}\right)$

References

Here are some references for further reading:

• "Trigonometry" by Michael Corral
• "Trigonometry for Dummies" by Mary Jane Sterling
• "Trigonometry: A Unit Circle Approach" by Charles P. McKeague

Conclusion

Introduction

In our previous article, we discussed how to evaluate a specific trigonometric expression using the given identities. In this article, we will provide a Q&A guide to help you understand the concepts and techniques involved in evaluating trigonometric expressions.

Q: What are the most common trigonometric identities?

A: The most common trigonometric identities include:

• $\sin^2 x + \cos^2 x = 1$
• $\tan x = \frac{\sin x}{\cos x}$
• $\cot x = \frac{\cos x}{\sin x}$
• $\sec x = \frac{1}{\cos x}$
• $\csc x = \frac{1}{\sin x}$

Q: How do I simplify a trigonometric expression?

A: To simplify a trigonometric expression, follow these steps:

1. Use the given identities to simplify the expression.
2. Perform the arithmetic operations in the correct order.
3. Simplify the expression by combining like terms.
4. Rationalize the denominator if necessary.

Q: What is the difference between $\sin x$ and $\cos x$?

A: $\sin x$ and $\cos x$ are two fundamental trigonometric functions. $\sin x$ represents the ratio of the length of the side opposite the angle to the length of the hypotenuse, while $\cos x$ represents the ratio of the length of the side adjacent to the angle to the length of the hypotenuse.

Q: How do I evaluate a trigonometric expression with multiple terms?

A: To evaluate a trigonometric expression with multiple terms, follow these steps:

1. Use the given identities to simplify each term.
2. Perform the arithmetic operations in the correct order.
3. Simplify the expression by combining like terms.
4. Rationalize the denominator if necessary.

Q: What is the significance of the unit circle in trigonometry?

A: The unit circle is a fundamental concept in trigonometry. It represents a circle with a radius of 1, centered at the origin of the coordinate plane. The unit circle is used to define the values of the trigonometric functions, such as $\sin x$ and $\cos x$.

Q: How do I use the unit circle to evaluate trigonometric expressions?

A: To use the unit circle to evaluate trigonometric expressions, follow these steps:

1. Identify the angle in question.
2. Determine the coordinates of the point on the unit circle corresponding to the angle.
3. Use the coordinates to evaluate the trigonometric expression.

Q: What are some common mistakes to avoid when evaluating trigonometric expressions?

A: Some common mistakes to avoid when evaluating trigonometric expressions include:

• Not using the given identities to simplify the expression.
• Performing the arithmetic operations in the incorrect order.
• Not simplifying the expression by combining like terms.
• Not rationalizing the denominator if necessary.

Q: How can I practice evaluating trigonometric expressions?

A: To practice evaluating trigonometric expressions, try the following:

• Work through practice problems in a textbook or online resource.
• Use online calculators or software to evaluate trigonometric expressions.
• Create your own practice problems and try to solve them.

Conclusion

In conclusion, evaluating trigonometric expressions is an essential skill in mathematics. By understanding the concepts and techniques involved, you can simplify complex expressions and solve problems with confidence. This Q&A guide provides a comprehensive overview of the topics and techniques involved in evaluating trigonometric expressions.