The sine of pi/4 (sin(π/4)) is 1/√2, which is approximately 0.7071.
Understanding sin(π/4)
Pi/4 radians is equivalent to 45 degrees. The sine function, in the context of the unit circle, represents the y-coordinate of a point on the unit circle corresponding to a given angle.
Derivation of sin(π/4)
We can derive this value using a 45-45-90 triangle. In such a triangle, the two legs are equal in length. Let's assume the legs have a length of 1. By the Pythagorean theorem, the hypotenuse has a length of √2.
- Sine (sin) is defined as the ratio of the opposite side to the hypotenuse.
Therefore, sin(45°) = sin(π/4) = opposite/hypotenuse = 1/√2.
Rationalizing the Denominator
It's common practice to rationalize the denominator, meaning we eliminate the square root from the denominator. We achieve this by multiplying both the numerator and denominator by √2:
(1/√2) * (√2/√2) = √2/2
So, sin(π/4) can also be expressed as √2/2, which is still approximately 0.7071.
Summary
Sin(π/4) is a fundamental trigonometric value. It equals 1/√2 or, equivalently, √2/2, which is approximately 0.7071. This value is derived from the geometry of a 45-45-90 triangle.
