Expressing the Answer with a Rational Denominator

Oct 13, 2025 by windailys.com 10 views

Introduction

In trigonometry, the tangent function is a fundamental concept that plays a crucial role in solving various mathematical problems. The tangent of an angle in a right-angled triangle is defined as the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle. In this article, we will focus on finding the exact value of tan(π/4), expressing our answer with a rational denominator.

Understanding the Problem

To find the exact value of tan(π/4), we need to recall the definition of the tangent function. The tangent of an angle θ is given by:

tan(θ) = sin(θ) / cos(θ)

where sin(θ) is the sine of the angle and cos(θ) is the cosine of the angle.

Recalling the Values of sin(π/4) and cos(π/4)

We know that sin(π/4) = cos(π/4) = 1/√2. This is because the sine and cosine functions are equal at an angle of π/4 (or 45 degrees).

Finding the Value of tan(π/4)

Now that we have the values of sin(π/4) and cos(π/4), we can substitute them into the formula for the tangent function:

tan(π/4) = sin(π/4) / cos(π/4) = (1/√2) / (1/√2) = 1

Expressing the Answer with a Rational Denominator

However, we are asked to express our answer with a rational denominator. To do this, we can multiply both the numerator and denominator by √2:

tan(π/4) = 1 / (1/√2) = (√2) / (√2/√2) = (√2) / 1 = √2

Conclusion

In this article, we have found the exact value of tan(π/4) and expressed our answer with a rational denominator. We have used the definition of the tangent function and the values of sin(π/4) and cos(π/4) to derive the result. The final answer is √2.

Additional Information

The value of tan(π/4) is equal to 1.
The value of tan(π/4) can be expressed with a rational denominator as √2.
The tangent function is a fundamental concept in trigonometry and is used to solve various mathematical problems.

Real-World Applications

The tangent function has many real-world applications, including:

Navigation: The tangent function is used in navigation to calculate the direction of an object.
Physics: The tangent function is used in physics to calculate the angle of incidence and reflection of light.
Engineering: The tangent function is used in engineering to calculate the angle of a slope or a ramp.

Common Mistakes

Not recalling the values of sin(π/4) and cos(π/4): It is essential to recall the values of sin(π/4) and cos(π/4) to find the value of tan(π/4).
Not expressing the answer with a rational denominator: It is crucial to express the answer with a rational denominator to meet the requirements of the problem.

Practice Problems

Find the value of tan(π/3).
Find the value of tan(π/6).
Find the value of tan(π/2).

Solutions

tan(π/3) = √3
tan(π/6) = 1/√3
tan(π/2) = undefined

Conclusion

Frequently Asked Questions

Q: What is the value of tan(π/4)?

A: The value of tan(π/4) is 1.

Q: Why is the value of tan(π/4) equal to 1?

A: The value of tan(π/4) is equal to 1 because the sine and cosine functions are equal at an angle of π/4 (or 45 degrees). Therefore, the ratio of the sine to the cosine is equal to 1.

Q: Can the value of tan(π/4) be expressed with a rational denominator?

A: Yes, the value of tan(π/4) can be expressed with a rational denominator as √2.

Q: How do you find the value of tan(π/4)?

A: To find the value of tan(π/4), you need to recall the definition of the tangent function and the values of sin(π/4) and cos(π/4). The tangent function is defined as the ratio of the sine to the cosine, so you can substitute the values of sin(π/4) and cos(π/4) into the formula.

Q: What are the real-world applications of the tangent function?

A: The tangent function has many real-world applications, including navigation, physics, and engineering. In navigation, the tangent function is used to calculate the direction of an object. In physics, the tangent function is used to calculate the angle of incidence and reflection of light. In engineering, the tangent function is used to calculate the angle of a slope or a ramp.

Q: What are some common mistakes to avoid when finding the value of tan(π/4)?

A: Some common mistakes to avoid when finding the value of tan(π/4) include not recalling the values of sin(π/4) and cos(π/4), and not expressing the answer with a rational denominator.

Q: Can you provide some practice problems for finding the value of tan(π/4)?

A: Yes, here are some practice problems for finding the value of tan(π/4):

Find the value of tan(π/3).
Find the value of tan(π/6).
Find the value of tan(π/2).

Solutions

tan(π/3) = √3
tan(π/6) = 1/√3
tan(π/2) = undefined

Additional Tips

Make sure to recall the values of sin(π/4) and cos(π/4) when finding the value of tan(π/4).
Express the answer with a rational denominator to meet the requirements of the problem.
Use the definition of the tangent function to find the value of tan(π/4).

Conclusion

In this article, we have provided a Q&A section for finding the value of tan(π/4). We have answered frequently asked questions, provided practice problems, and offered additional tips for finding the value of tan(π/4). We hope this article has been helpful in understanding the concept of the tangent function and finding the value of tan(π/4).