How do you use the half angle formula to find sin(67.5) ?

To find sin(67.5°) using the half-angle formula, we can follow these steps:

Step 1: Recognize the half-angle identity
The half-angle identity for sine is given by:

sin(θ/2) = ±√[(1 - cos(θ)) / 2]

Here, 67.5° is half of 135°. So, we can use the half-angle identity with θ = 135°.

Step 2: Apply the half-angle formula
To find sin(67.5°), we use:

sin(67.5°) = √[(1 - cos(135°)) / 2]

Step 3: Find cos(135°)
We know that:

cos(135°) = -√2 / 2

This is because 135° is in the second quadrant, and the cosine of 135° is negative.

Step 4: Substitute the value of cos(135°) into the formula
Now, substitute cos(135°) = -√2 / 2 into the formula:

sin(67.5°) = √[(1 - (-√2 / 2)) / 2]

sin(67.5°) = √[(1 + √2 / 2) / 2]

Step 5: Simplify the expression
To simplify the expression:

sin(67.5°) = √[(2/2 + √2 / 2) / 2] sin(67.5°) = √[(2 + √2) / 4]

Step 6: Final simplification
Now, simplify further:

sin(67.5°) = √[(2 + √2) / 4] sin(67.5°) = √(2 + √2) / 2

This is the exact value of sin(67.5°).

Answer:
sin(67.5°) = √(2 + √2) / 2