Calculating the Tangent of 45 Degrees

Oct 13, 2025 by magicalceylontours.com 143 views

The Side Lengths of a 45-45-90 Triangle and the Tangent of 45 Degrees

In this article, we will explore the relationship between the side lengths of a 45-45-90 triangle and the tangent of 45 degrees. A 45-45-90 triangle is a special right triangle with two equal legs and a hypotenuse that is equal to the square root of two times the length of a leg. The tangent of an angle in a right 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.

The Side Lengths of a 45-45-90 Triangle

The side lengths of a 45-45-90 triangle are in the ratio $1: 1: \sqrt{2}$ . This means that if the length of one leg is $x$ , then the length of the other leg is also $x$ , and the length of the hypotenuse is $x\sqrt{2}$ . This ratio is a fundamental property of 45-45-90 triangles and is used to solve problems involving these triangles.

The Tangent of 45 Degrees

The tangent of an angle in a right 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 angle of interest is 45 degrees. Since the triangle is isosceles, the side opposite the 45-degree angle is equal to the side adjacent to the 45-degree angle. Therefore, the tangent of 45 degrees is equal to the ratio of the length of the side opposite the 45-degree angle to the length of the side adjacent to the 45-degree angle.

Calculating the Tangent of 45 Degrees

To calculate the tangent of 45 degrees, we need to find the ratio of the length of the side opposite the 45-degree angle to the length of the side adjacent to the 45-degree angle. Since the side lengths of a 45-45-90 triangle are in the ratio $1: 1: \sqrt{2}$ , we can let the length of one leg be $x$ . Then, the length of the other leg is also $x$ , and the length of the hypotenuse is $x\sqrt{2}$ . The side opposite the 45-degree angle is one of the legs, and the side adjacent to the 45-degree angle is also one of the legs. Therefore, the tangent of 45 degrees is equal to the ratio of the length of one leg to the length of the other leg.

Simplifying the Tangent of 45 Degrees

Since the length of one leg is $x$ and the length of the other leg is also $x$ , the ratio of the length of one leg to the length of the other leg is equal to 1. Therefore, the tangent of 45 degrees is equal to 1.

In conclusion, the tangent of 45 degrees is equal to 1. This is because the side lengths of a 45-45-90 triangle are in the ratio $1: 1: \sqrt{2}$ , and the tangent of an angle in a right 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. Since the side opposite the 45-degree angle is equal to the side adjacent to the 45-degree angle, the tangent of 45 degrees is equal to 1.

The correct answer is C. 1.

"Geometry: A Comprehensive Introduction" by Dan Pedoe
"Trigonometry: A Unit Circle Approach" by Charles P. McKeague
"Mathematics for the Nonmathematician" by Morris Kline

What is the tangent of 45 degrees? Is it equal to $\sqrt{2}$ , $\frac{1}{2}$ , 1, or $\frac{\sqrt{2}}{2}$ ? Let us know your thoughts in the comments below!
The Side Lengths of a 45-45-90 Triangle and the Tangent of 45 Degrees: Q&A

In our previous article, we explored the relationship between the side lengths of a 45-45-90 triangle and the tangent of 45 degrees. We discovered that the tangent of 45 degrees is equal to 1. In this article, we will answer some frequently asked questions about the tangent of 45 degrees and the side lengths of a 45-45-90 triangle.

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

A: The ratio of the side lengths of a 45-45-90 triangle is $1: 1: \sqrt{2}$ . This means that if the length of one leg is $x$ , then the length of the other leg is also $x$ , and the length of the hypotenuse is $x\sqrt{2}$ .

Q: What is the tangent of 45 degrees?

A: The tangent of 45 degrees is equal to 1. This is because the side opposite the 45-degree angle is equal to the side adjacent to the 45-degree angle.

Q: How do you calculate the tangent of 45 degrees?

A: To calculate the tangent of 45 degrees, you need to find the ratio of the length of the side opposite the 45-degree angle to the length of the side adjacent to the 45-degree angle. Since the side lengths of a 45-45-90 triangle are in the ratio $1: 1: \sqrt{2}$ , you can let the length of one leg be $x$ . Then, the length of the other leg is also $x$ , and the length of the hypotenuse is $x\sqrt{2}$ . The side opposite the 45-degree angle is one of the legs, and the side adjacent to the 45-degree angle is also one of the legs. Therefore, the tangent of 45 degrees is equal to the ratio of the length of one leg to the length of the other leg.

Q: What is the relationship between the tangent of 45 degrees and the side lengths of a 45-45-90 triangle?

A: The tangent of 45 degrees is equal to the ratio of the length of the side opposite the 45-degree angle to the length of the side adjacent to the 45-degree angle. Since the side lengths of a 45-45-90 triangle are in the ratio $1: 1: \sqrt{2}$ , the tangent of 45 degrees is equal to 1.

Q: Can you give an example of a 45-45-90 triangle?

A: Yes, here is an example of a 45-45-90 triangle:

Leg 1: 3 inches
Leg 2: 3 inches
Hypotenuse: $3\sqrt{2}$ inches

In this example, the length of one leg is 3 inches, and the length of the other leg is also 3 inches. The length of the hypotenuse is $3\sqrt{2}$ inches.

Q: What are some common mistakes to avoid when working with 45-45-90 triangles?

A: Here are some common mistakes to avoid when working with 45-45-90 triangles:

Not using the correct ratio of side lengths
Not using the correct formula for the tangent of 45 degrees
Not checking your work for errors

In conclusion, the tangent of 45 degrees is equal to 1. We hope that this article has helped to clarify the relationship between the tangent of 45 degrees and the side lengths of a 45-45-90 triangle. If you have any further questions, please don't hesitate to ask.

"Geometry: A Comprehensive Introduction" by Dan Pedoe
"Trigonometry: A Unit Circle Approach" by Charles P. McKeague
"Mathematics for the Nonmathematician" by Morris Kline

Do you have any questions about the tangent of 45 degrees or the side lengths of a 45-45-90 triangle? Let us know in the comments below!