To find the exact value of sin 225°, we can utilize the half-angle formula. The half-angle formula for sine is given by:
sin(θ/2) = ±√((1 – cos(θ)) / 2)
In this case, we want to express 225° as half of another angle. We know that 225° is half of 450°:
- θ = 450°
First, we need to find cos(450°). The cosine function has a period of 360°, so:
- cos(450°) = cos(450° – 360°) = cos(90°) = 0
Now we can plug that into the half-angle formula:
sin(225°) = sin(450° / 2) = √((1 – cos(450°)) / 2) = √((1 – 0) / 2) = √(1/2) = √2 / 2
Since 225° is in the third quadrant where sine is negative, we take the negative value:
So, sin(225°) = -√2 / 2.
This value represents the exact sine of 225°, and the explanation provided utilizes the half-angle formula effectively.