## cos π/4 (45)

### The expression “cos π/4” represents the cosine function evaluated at π/4 radians, which is equivalent to 45 degrees

The expression “cos π/4” represents the cosine function evaluated at π/4 radians, which is equivalent to 45 degrees. Before we can find the value of cos π/4, let’s have a look at the unit circle.

The unit circle is a circle with a radius of 1, centered at the origin of the coordinate plane. It is commonly used in trigonometry to represent angles and to calculate trigonometric functions such as sine, cosine, and tangent.

To find the value of cos π/4, we look at the point on the unit circle corresponding to π/4 radians or 45 degrees. At this angle, the x-coordinate of the point on the unit circle represents cos π/4.

Drawing a line from the point on the unit circle at π/4 to the x-axis, we can see that it forms an isosceles right triangle. The hypotenuse of the triangle (line segment from the origin to the point) has a length 1, as it is the radius of the unit circle. The two sides of the triangle have the same length, forming a right angle at the origin.

Using the Pythagorean theorem, we can solve for the lengths of the sides of the triangle. Let’s call one of the sides “a” and denote the other side as “b”. Since it is an isosceles right triangle, a = b.

By Pythagorean theorem:

a^2 + b^2 = hypotenuse^2

a^2 + a^2 = 1^2

2a^2 = 1

a^2 = 1/2

a = √(1/2)

Since the triangle is in the first quadrant, both a and b are positive. Therefore, a = b = √(1/2) = 1/√2.

Now, to find the cosine of π/4, we look at the x-coordinate of the point on the unit circle, which is the side “a” of the triangle.

cos π/4 = a = 1/√2

But dividing by √2 in both the numerator and the denominator, we get:

cos π/4 = 1/√2 * (√2/√2) = √2/2

Therefore, the value of cos π/4 (or cos 45°) is √2/2.

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