To find the exact value of
cos 225 degrees using the half-angle formula, we can break it down into simple steps. The half-angle formula for cosine is given by:
cos(θ/2) = ± √((1 + cos θ)/2)
In our case, since
225 degrees is equivalent to
2 * 112.5 degrees, we can find
cos 225 degrees by using the half-angle formula for
cos 225 degrees = cos(450/2 degrees).
First, we determine
cos 450 degrees:
450 degrees is equivalent to
90 degrees (as 450 – 360 = 90). The cosine of 90 degrees is:
cos 90 degrees = 0
Next, we find cos 225 degrees using the angle 112.5 degrees, since:
cos 225 degrees = cos(180 + 45 degrees)
From trigonometric principles, we can deduce that:
cos(180 + θ) = -cos θ
Therefore:
cos 225 degrees = -cos 45 degrees
Since
cos 45 degrees = √2/2, we have:
cos 225 degrees = -√2/2
To summarize:
- Use the half-angle formula: cos(θ/2) = ± √((1 + cos θ)/2)
- Determine the relevant angle values and their cosines.
- In our case, cos 225 degrees is found to be: -√2/2.
Now you’ve successfully calculated the exact value of cos 225 degrees utilizing the half-angle formula!