Understanding the Cosine of 45 Degrees | Exploring the Unit Circle and Isosceles Right Triangle

cos 45°

The cosine of 45 degrees is equal to √2/2

The cosine of 45 degrees is equal to √2/2.

To understand why, we need to refer to the unit circle. The unit circle is a circle with a radius of 1 unit, centered at the origin (0,0) in a coordinate plane. The angle measurement starts from the positive x-axis and moves counterclockwise.

At 45 degrees, we can draw a right-angled triangle inside the unit circle. The hypotenuse of this triangle intersects the circle at a point (x, y). The x-coordinate of that point is the cosine of the angle.

In the case of 45 degrees, the triangle is an isosceles right triangle with two legs of equal length. Since the hypotenuse is the radius of the unit circle (which is 1 unit), both legs of the triangle are √2/2.

Using the Pythagorean theorem, we can find the length of the legs:
(a^2 + a^2) = (1^2)
2a^2 = 1
a^2 = 1/2
a = √(1/2)
a = √2/2

Therefore, the x-coordinate, which is the cosine of 45 degrees, is also √2/2.

Hence, cos 45° = √2/2.

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