The Pythagorean Theorem and Trigonometry

Oct 13, 2025 by magicalceylontours.com 75 views

Introduction

In mathematics, particularly in geometry and trigonometry, errors can be subtle and lead to incorrect conclusions. Tavon, a student, is tasked with finding the value of EF using the Pythagorean theorem and trigonometric functions. However, he makes a critical mistake that affects the accuracy of his calculation. In this article, we will identify the error Tavon makes and provide a step-by-step solution to find the correct value of EF.

The Pythagorean Theorem and Trigonometry

The Pythagorean theorem states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. Mathematically, this can be expressed as:

a^2 + b^2 = c^2

where a and b are the lengths of the two sides, and c is the length of the hypotenuse.

In the given problem, Tavon is asked to find EF, which is the length of the hypotenuse of a right-angled triangle with sides DE and DF. Using the Pythagorean theorem, we can write:

EF^2 = DE^2 + DF^2

However, Tavon also uses the cosine function to calculate the value of EF. The cosine function is defined as:

cos(x) = adjacent side / hypotenuse

In this case, Tavon uses the cosine function to calculate the value of EF, but he makes a mistake in his calculation.

Tavon's Calculation

Tavon's calculation is as follows:

EF^2 = 8^2 + 10^2 - 2(8)(10) cos 44°

EF^2 ≈ 64 + 100 - 160 × 0.7193

EF ≈ √(164 - 115.072)

EF ≈ √48.928

EF ≈ 7.00

The Error in Tavon's Calculation

The error in Tavon's calculation is that he uses the cosine function to calculate the value of EF, but he does not take into account the fact that the cosine function is defined as the ratio of the adjacent side to the hypotenuse. In this case, the cosine function is used to calculate the value of EF, but the value of EF is not known.

To correct this error, we need to use the Pythagorean theorem to calculate the value of EF. We can do this by using the following formula:

EF = √(DE^2 + DF^2)

Substituting the values of DE and DF, we get:

EF = √(8^2 + 10^2)

EF = √(64 + 100)

EF = √164

EF ≈ 12.81

Conclusion

In conclusion, Tavon makes a mistake in his calculation by using the cosine function to calculate the value of EF. To correct this error, we need to use the Pythagorean theorem to calculate the value of EF. The correct value of EF is approximately 12.81.

Discussion

The discussion category for this problem is mathematics, specifically geometry and trigonometry. The problem requires the use of the Pythagorean theorem and trigonometric functions to calculate the value of EF.

Key Takeaways

The Pythagorean theorem is used to calculate the value of EF in a right-angled triangle.
The cosine function is used to calculate the value of EF, but it requires the value of EF to be known.
The correct value of EF is calculated using the Pythagorean theorem.

Recommendations

Students should be careful when using trigonometric functions to calculate values in a right-angled triangle.
The Pythagorean theorem should be used to calculate the value of EF in a right-angled triangle.
Students should check their calculations carefully to avoid errors.

Glossary

Pythagorean theorem: a mathematical formula used to calculate the value of EF in a right-angled triangle.
Trigonometric functions: mathematical functions used to calculate values in a right-angled triangle.
Hypotenuse: the side opposite the right angle in a right-angled triangle.
Adjacent side: the side next to the angle in a right-angled triangle.
Tavon's Mistake in Calculating EF: Q&A =====================================

Introduction

In our previous article, we discussed Tavon's mistake in calculating the value of EF using the Pythagorean theorem and trigonometric functions. We identified the error in his calculation and provided a step-by-step solution to find the correct value of EF. In this article, we will answer some frequently asked questions related to Tavon's mistake and provide additional insights into the problem.

Q&A

Q: What is the Pythagorean theorem?

A: The Pythagorean theorem is a mathematical formula used to calculate the value of EF in a right-angled triangle. It states that the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.

Q: What is the cosine function?

A: The cosine function is a trigonometric function used to calculate the value of EF in a right-angled triangle. It is defined as the ratio of the adjacent side to the hypotenuse.

Q: Why did Tavon make a mistake in his calculation?

A: Tavon made a mistake in his calculation because he used the cosine function to calculate the value of EF, but he did not take into account the fact that the cosine function is defined as the ratio of the adjacent side to the hypotenuse. This led to an incorrect calculation of the value of EF.

Q: How can we correct Tavon's mistake?

A: To correct Tavon's mistake, we need to use the Pythagorean theorem to calculate the value of EF. We can do this by using the following formula:

EF = √(DE^2 + DF^2)

Substituting the values of DE and DF, we get:

EF = √(8^2 + 10^2)

EF = √(64 + 100)

EF = √164

EF ≈ 12.81

Q: What is the correct value of EF?

A: The correct value of EF is approximately 12.81.

Q: Why is it important to use the Pythagorean theorem to calculate the value of EF?

A: It is important to use the Pythagorean theorem to calculate the value of EF because it provides an accurate calculation of the value of EF. The cosine function can be used to calculate the value of EF, but it requires the value of EF to be known, which can lead to an incorrect calculation.

Q: What are some common mistakes to avoid when calculating the value of EF?

A: Some common mistakes to avoid when calculating the value of EF include:

Using the cosine function to calculate the value of EF without taking into account the fact that the cosine function is defined as the ratio of the adjacent side to the hypotenuse.
Not using the Pythagorean theorem to calculate the value of EF.
Not checking calculations carefully to avoid errors.