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)
Now, 225 degrees is half of 450 degrees, so we can set:
θ = 450°
Next, we will find the cosine of 450 degrees. Since 450 degrees is equivalent to 450 – 360 = 90 degrees, we have:
cos(450°) = cos(90°) = 0
Now, we can substitute this value into the half-angle formula:
sin(225°) = sin(450°/2) = sin(225°) = ±√((1 – cos(450)) / 2)
Substituting the cosine value:
sin(225°) = ±√((1 – 0) / 2) = ±√(1 / 2)
This simplifies to:
sin(225°) = ±√(0.5) = ±√(1)/√(2) = ±1/√(2) = ±√2 / 2
To determine the correct sign, we note that 225 degrees lies in the third quadrant, where the sine function is negative. Thus, we take the negative value:
sin(225°) = -√2 / 2
In summary, using the half-angle formula, we find that the exact value of sin 225° is -√2 / 2.