Evaluate The Expression: $\[ 3 \cot^2 60^{\circ} + \sec^2 45^{\circ} = \\]

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Introduction

In this article, we will delve into the world of trigonometry and evaluate the given expression: 3cot^2 60^{\circ} + sec^2 45^{\circ}. We will break down the expression into its individual components, apply the relevant trigonometric identities, and simplify the expression to obtain the final result.

Understanding the Components

Before we begin, let's understand the components of the given expression.

  • cot: The cotangent function is the reciprocal of the tangent function. It is defined as cot(x) = 1/tan(x).
  • sec: The secant function is the reciprocal of the cosine function. It is defined as sec(x) = 1/cos(x).
  • 60^{\circ}: This is an angle in degrees.
  • 45^{\circ}: This is an angle in degrees.

Evaluating the Expression

Now that we have a good understanding of the components, let's evaluate the expression.

Evaluating cot^2 60^{\circ}

To evaluate cot^2 60^{\circ}, we need to find the value of cot 60^{\circ} first.

  • **cot 60^\circ}** Since 60^{\circ is a special angle, we know that cot 60^{\circ} = 1/√3.
  • **cot^2 60^\circ}** Now that we have the value of cot 60^{\circ, we can square it to get cot^2 60^{\circ} = (1/√3)^2 = 1/3.

Evaluating sec^2 45^{\circ}

To evaluate sec^2 45^{\circ}, we need to find the value of sec 45^{\circ} first.

  • **sec 45^\circ}** Since 45^{\circ is a special angle, we know that sec 45^{\circ} = 1/√2.
  • **sec^2 45^\circ}** Now that we have the value of sec 45^{\circ, we can square it to get sec^2 45^{\circ} = (1/√2)^2 = 1/2.

Evaluating the Expression

Now that we have the values of cot^2 60^{\circ} and sec^2 45^{\circ}, we can substitute them into the original expression.

  • 3cot^2 60^{\circ} + sec^2 45^{\circ}: Substituting the values we found earlier, we get 3(1/3) + 1/2 = 1 + 1/2 = 3/2.

Conclusion

In this article, we evaluated the expression 3cot^2 60^{\circ} + sec^2 45^{\circ} by breaking it down into its individual components, applying the relevant trigonometric identities, and simplifying the expression to obtain the final result. We found that the expression simplifies to 3/2.

Frequently Asked Questions

  • What is the value of cot 60^{\circ}?
    • The value of cot 60^{\circ} is 1/√3.
  • What is the value of sec 45^{\circ}?
    • The value of sec 45^{\circ} is 1/√2.
  • What is the value of cot^2 60^{\circ}?
    • The value of cot^2 60^{\circ} is 1/3.
  • What is the value of sec^2 45^{\circ}?
    • The value of sec^2 45^{\circ} is 1/2.
  • What is the final value of the expression 3cot^2 60^{\circ} + sec^2 45^{\circ}?
    • The final value of the expression 3cot^2 60^{\circ} + sec^2 45^{\circ} is 3/2.

References

  • Trigonometry: A branch of mathematics that deals with the relationships between the sides and angles of triangles.
  • cotangent: The reciprocal of the tangent function.
  • secant: The reciprocal of the cosine function.
  • 60^{\circ}: A special angle in degrees.
  • 45^{\circ}: A special angle in degrees.

Further Reading

  • Trigonometry for Dummies: A comprehensive guide to trigonometry for beginners.
  • Trigonometric Identities: A list of trigonometric identities and their proofs.
  • Trigonometry Formulas: A list of trigonometry formulas and their applications.

Conclusion

In this article, we evaluated the expression 3cot^2 60^{\circ} + sec^2 45^{\circ} by breaking it down into its individual components, applying the relevant trigonometric identities, and simplifying the expression to obtain the final result. We found that the expression simplifies to 3/2.

Introduction

In our previous article, we evaluated the expression 3cot^2 60^{\circ} + sec^2 45^{\circ} and found that it simplifies to 3/2. In this article, we will answer some frequently asked questions related to the expression and provide additional information to help readers understand the concept better.

Q&A

Q1: What is the value of cot 60^{\circ}?

A1: The value of cot 60^{\circ} is 1/√3.

Q2: What is the value of sec 45^{\circ}?

A2: The value of sec 45^{\circ} is 1/√2.

Q3: What is the value of cot^2 60^{\circ}?

A3: The value of cot^2 60^{\circ} is 1/3.

Q4: What is the value of sec^2 45^{\circ}?

A4: The value of sec^2 45^{\circ} is 1/2.

Q5: What is the final value of the expression 3cot^2 60^{\circ} + sec^2 45^{\circ}?

A5: The final value of the expression 3cot^2 60^{\circ} + sec^2 45^{\circ} is 3/2.

Q6: How do I evaluate the expression 3cot^2 60^{\circ} + sec^2 45^{\circ}?

A6: To evaluate the expression 3cot^2 60^{\circ} + sec^2 45^{\circ}, you need to follow these steps:

  1. Evaluate cot 60^{\circ} and sec 45^{\circ}.
  2. Square the values of cot 60^{\circ} and sec 45^{\circ} to get cot^2 60^{\circ} and sec^2 45^{\circ}.
  3. Multiply cot^2 60^{\circ} by 3.
  4. Add sec^2 45^{\circ} to the result.
  5. Simplify the expression to get the final value.

Q7: What are the trigonometric identities used in this expression?

A7: The trigonometric identities used in this expression are:

  • cot(x) = 1/tan(x)
  • sec(x) = 1/cos(x)
  • cot^2(x) = 1/cos^2(x) - 1
  • sec^2(x) = 1 + tan^2(x)

Q8: How do I apply the trigonometric identities in this expression?

A8: To apply the trigonometric identities in this expression, you need to follow these steps:

  1. Identify the trigonometric functions in the expression.
  2. Apply the relevant trigonometric identities to simplify the expression.
  3. Simplify the expression further to get the final value.

Q9: What are the special angles used in this expression?

A9: The special angles used in this expression are:

  • 60^{\circ}
  • 45^{\circ}

Q10: How do I evaluate the expression 3cot^2 60^{\circ} + sec^2 45^{\circ} for other angles?

A10: To evaluate the expression 3cot^2 60^{\circ} + sec^2 45^{\circ} for other angles, you need to follow these steps:

  1. Evaluate cot and sec for the given angle.
  2. Square the values of cot and sec to get cot^2 and sec^2.
  3. Multiply cot^2 by 3.
  4. Add sec^2 to the result.
  5. Simplify the expression to get the final value.

Conclusion

In this article, we answered some frequently asked questions related to the expression 3cot^2 60^{\circ} + sec^2 45^{\circ} and provided additional information to help readers understand the concept better. We hope that this article has been helpful in clarifying any doubts that readers may have had.

Frequently Asked Questions

  • What is the value of cot 60^{\circ}?
    • The value of cot 60^{\circ} is 1/√3.
  • What is the value of sec 45^{\circ}?
    • The value of sec 45^{\circ} is 1/√2.
  • What is the value of cot^2 60^{\circ}?
    • The value of cot^2 60^{\circ} is 1/3.
  • What is the value of sec^2 45^{\circ}?
    • The value of sec^2 45^{\circ} is 1/2.
  • What is the final value of the expression 3cot^2 60^{\circ} + sec^2 45^{\circ}?
    • The final value of the expression 3cot^2 60^{\circ} + sec^2 45^{\circ} is 3/2.

References

  • Trigonometry: A branch of mathematics that deals with the relationships between the sides and angles of triangles.
  • cotangent: The reciprocal of the tangent function.
  • secant: The reciprocal of the cosine function.
  • 60^{\circ}: A special angle in degrees.
  • 45^{\circ}: A special angle in degrees.

Further Reading

  • Trigonometry for Dummies: A comprehensive guide to trigonometry for beginners.
  • Trigonometric Identities: A list of trigonometric identities and their proofs.
  • Trigonometry Formulas: A list of trigonometry formulas and their applications.

Conclusion

In this article, we answered some frequently asked questions related to the expression 3cot^2 60^{\circ} + sec^2 45^{\circ} and provided additional information to help readers understand the concept better. We hope that this article has been helpful in clarifying any doubts that readers may have had.