Exploring the Cosine of 45 Degrees | Calculation, Triangle Labeling, and Rationalization

cos 45°

The cosine of 45 degrees, written as cos 45°, is a trigonometric function that calculates the ratio of the length of the adjacent side to the hypotenuse in a right triangle with a 45-degree angle

The cosine of 45 degrees, written as cos 45°, is a trigonometric function that calculates the ratio of the length of the adjacent side to the hypotenuse in a right triangle with a 45-degree angle.

To understand how to calculate it, we first need to label the sides of the triangle. Let’s assume that the hypotenuse (the side opposite the right angle) has a length of 1 unit. Since the triangle has a 45-degree angle, the other two sides, adjacent and opposite to the angle, will have the same length.

Using the Pythagorean theorem, we can calculate the length of the other two sides. Let’s denote the length of the adjacent and opposite sides as “a” units. According to the theorem, we have:

a^2 + a^2 = 1^2
2a^2 = 1
a^2 = 1/2
a = sqrt(1/2) = 1/sqrt(2)

Now that we know the values of the length of the adjacent side and the hypotenuse, we can calculate the cosine of 45 degrees. The cosine function is defined as:

cos θ = adjacent/hypotenuse

In this case, θ = 45 degrees, so we have:

cos 45° = (1/sqrt(2))/1 = 1/sqrt(2)

To rationalize the denominator, we need to multiply both the numerator and denominator by sqrt(2):

cos 45° = (1/sqrt(2)) * (sqrt(2)/sqrt(2)) = sqrt(2)/2

So, the cosine of 45 degrees is equal to sqrt(2)/2, which is an irrational number approximately equal to 0.7071.

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