Find The Exact Value Of 1) Sec 45° Tan 30° -Sin 60°​

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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 finding the exact value of a given trigonometric expression, which involves the use of various trigonometric identities and formulas.

The Given Expression

The given expression is:

Sec 45° tan 30° - Sin 60°

To find the exact value of this expression, we need to simplify it by using various trigonometric identities and formulas.

Step 1: Simplify the Expression

We can start by simplifying the expression using the following trigonometric identities:

  • Sec θ = 1 / cos θ
  • tan θ = sin θ / cos θ
  • sin (90° - θ) = cos θ

Using these identities, we can rewrite the expression as:

(1 / cos 45°) (sin 30° / cos 30°) - sin 60°

Step 2: Evaluate the Trigonometric Functions

Next, we need to evaluate the trigonometric functions in the expression. We know that:

  • sin 30° = 1/2
  • cos 30° = √3/2
  • sin 60° = √3/2
  • cos 45° = 1/√2

Substituting these values into the expression, we get:

(1 / (1/√2)) ((1/2) / (√3/2)) - (√3/2)

Step 3: Simplify the Expression

Now, we can simplify the expression by canceling out the common factors:

√2 ((1/2) / (√3/2)) - (√3/2)

Using the property of fractions, we can rewrite the expression as:

(√2 / 2) - (√3/2)

Step 4: Find a Common Denominator

To add or subtract fractions, we need to have a common denominator. In this case, the common denominator is 2. So, we can rewrite the expression as:

(√2 / 2) - (√3/2) = (√2 - √3) / 2

Conclusion

In this article, we have found the exact value of the given trigonometric expression, Sec 45° tan 30° - Sin 60°. We used various trigonometric identities and formulas to simplify the expression and evaluate the trigonometric functions. The final answer is (√2 - √3) / 2.

Trigonometric Identities Used

In this article, we used the following trigonometric identities:

  • Sec θ = 1 / cos θ
  • tan θ = sin θ / cos θ
  • sin (90° - θ) = cos θ

Trigonometric Functions Used

In this article, we used the following trigonometric functions:

  • sin 30° = 1/2
  • cos 30° = √3/2
  • sin 60° = √3/2
  • cos 45° = 1/√2

Tips and Tricks

When working with trigonometric expressions, it is essential to use the correct trigonometric identities and formulas. Additionally, make sure to evaluate the trigonometric functions correctly and simplify the expression step by step.

Real-World Applications

Trigonometry has numerous real-world applications, including:

  • Physics: Trigonometry is used to describe the motion of objects in terms of position, velocity, and acceleration.
  • Engineering: Trigonometry is used to design and analyze various engineering systems, such as bridges, buildings, and electrical circuits.
  • Navigation: Trigonometry is used to determine the position and direction of objects on the Earth's surface.

Conclusion

Introduction

In our previous article, we discussed how to find the exact value of a given trigonometric expression, Sec 45° tan 30° - Sin 60°. In this article, we will provide a Q&A section to help you better understand the concepts and formulas used in trigonometry.

Q&A

Q: What is the difference between secant and cosine?

A: The secant of an angle is the reciprocal of the cosine of that angle. In other words, sec θ = 1 / cos θ.

Q: How do I simplify a trigonometric expression?

A: To simplify a trigonometric expression, you need to use the correct trigonometric identities and formulas. Start by identifying the trigonometric functions involved and then use the identities to simplify the expression.

Q: What is the value of sin 45°?

A: The value of sin 45° is √2 / 2.

Q: How do I evaluate the trigonometric functions in a given expression?

A: To evaluate the trigonometric functions in a given expression, you need to use the values of the trigonometric functions for the specific angles involved. For example, sin 30° = 1/2, cos 30° = √3/2, sin 60° = √3/2, and cos 45° = 1/√2.

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

A: The sine and cosine functions are two of the most basic trigonometric functions. The sine function represents the ratio of the opposite side to the hypotenuse in a right triangle, while the cosine function represents the ratio of the adjacent side to the hypotenuse.

Q: How do I use the Pythagorean identity to simplify a trigonometric expression?

A: The Pythagorean identity states that sin^2 θ + cos^2 θ = 1. You can use this identity to simplify a trigonometric expression by substituting the values of the sine and cosine functions.

Q: What is the value of tan 45°?

A: The value of tan 45° is 1.

Q: How do I use the trigonometric identities to find the exact value of a trigonometric expression?

A: To find the exact value of a trigonometric expression, you need to use the correct trigonometric identities and formulas. Start by identifying the trigonometric functions involved and then use the identities to simplify the expression.

Q: What are some common trigonometric identities?

A: Some common trigonometric identities include:

  • Sec θ = 1 / cos θ
  • tan θ = sin θ / cos θ
  • sin (90° - θ) = cos θ
  • cos (90° - θ) = sin θ

Conclusion

In conclusion, finding the exact value of a trigonometric expression requires a thorough understanding of trigonometric identities and formulas. By using these identities and formulas, you can simplify the expression and evaluate the trigonometric functions to find the final answer. We hope this Q&A section has helped you better understand the concepts and formulas used in trigonometry.

Trigonometry Formulas and Identities

Here are some common trigonometry formulas and identities:

  • Pythagorean Identity: sin^2 θ + cos^2 θ = 1
  • Secant Identity: sec θ = 1 / cos θ
  • Tangent Identity: tan θ = sin θ / cos θ
  • Sine and Cosine Identities: sin (90° - θ) = cos θ, cos (90° - θ) = sin θ

Tips and Tricks

When working with trigonometric expressions, it is essential to use the correct trigonometric identities and formulas. Additionally, make sure to evaluate the trigonometric functions correctly and simplify the expression step by step.

Real-World Applications

Trigonometry has numerous real-world applications, including:

  • Physics: Trigonometry is used to describe the motion of objects in terms of position, velocity, and acceleration.
  • Engineering: Trigonometry is used to design and analyze various engineering systems, such as bridges, buildings, and electrical circuits.
  • Navigation: Trigonometry is used to determine the position and direction of objects on the Earth's surface.

Conclusion

In conclusion, trigonometry is a fundamental subject that has numerous applications in various fields. By understanding the concepts and formulas used in trigonometry, you can simplify trigonometric expressions and evaluate trigonometric functions to find the final answer. We hope this article has helped you better understand the concepts and formulas used in trigonometry.