# Understanding 45-45-90 Degrees Angles

• Miike012
In summary, the value of 1/(2)^(1/2) in reference to a 45-45-90 degree angle is the ratio of the length of the perpendicular to the length of the hypotenuse of the right angled triangle. This trigonometric ratio is independent of the size of the triangle and plays a crucial role in topics such as vector analysis, calculus, and Fourier analysis. Furthermore, it helps develop several other areas of mathematics beyond its initial definition as a ratio.

## Homework Statement

Say I am dealing with a 45-45-90 deg angle

then sin(45 deg) = 1/(2)^(1/2)

What exactly is the value of 1/(2)^(1/2) referring to?
is is the measure of a side or what?

## The Attempt at a Solution

naively, sine of an angle, is the ratio of the length of the perpendicular to the length of the hypotenuse of the right angled triangle that this angle creates.

So, in a 45-45-90 triangle, the sides are 'a', 'a' and 'sqrt(2)*a.

So according to the above definition, sin(45) = a/sqrt(2)*a = 1/sqrt(2)

I just don't see why the ratio of the lengths of a triangle would be beneficial... so once I find the sin, cos, tan, or their recipricols of an angle... how is that ratio going to help me?

Well, firstly we see that trigonometric ratios, even though defined with respect to a particular triangle, it is independent of the size of the triangle (in my example, 'a').

In fact, as you learn more, the dependency on a right angle triangle in defining sine and cos is completely removed. Given this, sines and cos retain the properties of the triangle without having anything to do it.

Trigonometric Functions have extreme importance in topics of vector analysis and calculus because of their very special properties. However, its major application comes in Fourier analysis, because of a theorem due to Fourier which states that

'every periodic function, can be written as a linear combination of sines and cosines.'

So you see, even though we begin with defining it as a ratio, it develops into something much more, which helps us develop several other areas of mathematics.

## 1. What is a 45-45-90 degree angle?

A 45-45-90 degree angle is a type of right triangle where the two shorter sides are equal in length and the longer side is equal to the square root of 2 times the length of the shorter sides. The three angles in this triangle measure 45 degrees, 45 degrees, and 90 degrees, respectively.

## 2. How do you identify a 45-45-90 degree angle?

A 45-45-90 degree angle can be identified by its unique side lengths. The two shorter sides will be equal in length, and the longer side will be the square root of 2 times the length of the shorter sides. Additionally, the angles can be measured using a protractor to confirm that they are 45 degrees, 45 degrees, and 90 degrees.

## 3. What is the Pythagorean Theorem and how does it relate to 45-45-90 degree angles?

The Pythagorean Theorem is a mathematical formula that states that in a right triangle, the square of the length of the hypotenuse (the longest side) is equal to the sum of the squares of the lengths of the other two sides. In a 45-45-90 degree angle, the length of the longer side is equal to the square root of 2 times the length of the shorter sides, which follows the Pythagorean Theorem.

## 4. How are 45-45-90 degree angles used in real life?

45-45-90 degree angles are commonly used in construction and engineering to create right angles and maintain symmetry in buildings and structures. They are also used in navigation and map-making to determine direction and distances between points. In mathematics, they are used to solve geometric problems and equations.

## 5. How can I calculate the missing side length in a 45-45-90 degree angle?

To calculate the missing side length in a 45-45-90 degree angle, you can use the Pythagorean Theorem or the special ratio of the sides (1:1:√2). For example, if the shorter sides are each 5 units in length, the longer side would be 5√2 units in length. To find a missing side, you can also use trigonometric functions such as sine, cosine, and tangent.