Finding the Tangent of 45 Degrees

Oct 13, 2025 by windailys.com 67 views

Introduction

In the world of geometry, triangles are fundamental shapes that have been studied for centuries. One of the most interesting types of triangles is the 45-45-90 triangle, which has side lengths in the ratio of 1:1:\sqrt{2}. This unique ratio makes the 45-45-90 triangle a special case in geometry, and its properties have been extensively studied. In this article, we will explore the side lengths of a 45-45-90 triangle and use this information to find the tangent of 45 degrees.

The Side Lengths of a 45-45-90 Triangle

A 45-45-90 triangle is a right-angled triangle with two equal acute angles, each measuring 45 degrees. The side lengths of a 45-45-90 triangle are in the ratio of 1:1:\sqrt{2}, where the two equal sides are the legs of the triangle, and the hypotenuse is the side opposite the right angle. This ratio is a fundamental property of 45-45-90 triangles and can be used to find the side lengths of the triangle.

The Tangent of 45 Degrees

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 the case of a 45-45-90 triangle, the two equal sides are the legs of the triangle, and the hypotenuse is the side opposite the right angle. Since the two equal sides are the legs of the triangle, the tangent of 45 degrees is equal to the ratio of the length of one leg to the length of the other leg.

Finding the Tangent of 45 Degrees

To find the tangent of 45 degrees, we can use the ratio of the side lengths of a 45-45-90 triangle. Let's assume that the length of one leg is 1 unit, and the length of the other leg is also 1 unit. The hypotenuse of the triangle is then equal to \sqrt{2} units. Using the definition of the tangent, we can write:

tan(45^{\circ}) = \frac{opposite}{adjacent} = \frac{1}{1} = 1

However, this is not the correct answer. We need to use the ratio of the side lengths to find the correct answer.

Using the Ratio of Side Lengths

Since the side lengths of a 45-45-90 triangle are in the ratio of 1:1:\sqrt{2}, we can write:

\frac{opposite}{adjacent} = \frac{1}{1} = \frac{\sqrt{2}}{2}

Therefore, the tangent of 45 degrees is equal to \frac{\sqrt{2}}{2}.

Conclusion

In conclusion, the side lengths of a 45-45-90 triangle are in the ratio of 1:1:\sqrt{2}. Using this ratio, we can find the tangent of 45 degrees, which is equal to \frac{\sqrt{2}}{2}. This result is consistent with the definition of the tangent and the properties of 45-45-90 triangles.

Final Answer

The final answer is:

A. \frac{\sqrt{2}}{2}

This answer is consistent with the definition of the tangent and the properties of 45-45-90 triangles.

Introduction

In our previous article, we explored the side lengths of a 45-45-90 triangle and used this information to find the tangent of 45 degrees. In this article, we will answer some of the most frequently asked questions about the side lengths of a 45-45-90 triangle and the tangent of 45 degrees.

Q: What is the ratio of the side lengths of a 45-45-90 triangle?

A: The side lengths of a 45-45-90 triangle are in the ratio of 1:1:\sqrt{2}, where the two equal sides are the legs of the triangle, and the hypotenuse is the side opposite the right angle.

Q: How do I find the side lengths of a 45-45-90 triangle?

A: To find the side lengths of a 45-45-90 triangle, you can use the ratio of 1:1:\sqrt{2}. Let's assume that the length of one leg is 1 unit, and the length of the other leg is also 1 unit. The hypotenuse of the triangle is then equal to \sqrt{2} units.

Q: What is the tangent of 45 degrees?

A: The tangent of 45 degrees is equal to the ratio of the length of one leg to the length of the other leg. Using the ratio of the side lengths, we can write:

tan(45^{\circ}) = \frac{opposite}{adjacent} = \frac{1}{1} = \frac{\sqrt{2}}{2}

Q: Why is the tangent of 45 degrees equal to \frac{\sqrt{2}}{2}?

A: The tangent of 45 degrees is equal to \frac\sqrt{2}}{2} because the side lengths of a 45-45-90 triangle are in the ratio of 11:\sqrt{2. This ratio is a fundamental property of 45-45-90 triangles and can be used to find the tangent of 45 degrees.

Q: Can I use the tangent of 45 degrees to find the side lengths of a 45-45-90 triangle?

A: No, you cannot use the tangent of 45 degrees to find the side lengths of a 45-45-90 triangle. The tangent of 45 degrees is a ratio of the side lengths, and it is used to find the tangent of an angle, not the side lengths of a triangle.

Q: What are some real-world applications of 45-45-90 triangles?

A: 45-45-90 triangles have many real-world applications, including:

Building design: 45-45-90 triangles are used in building design to create strong and stable structures.
Engineering: 45-45-90 triangles are used in engineering to create efficient and effective systems.
Art: 45-45-90 triangles are used in art to create balanced and harmonious compositions.

Q: Can I use a calculator to find the tangent of 45 degrees?

A: Yes, you can use a calculator to find the tangent of 45 degrees. However, it's always a good idea to understand the underlying math and the properties of 45-45-90 triangles.