How do you find the exact value of sin 67.5 degrees?

To find the exact value of sin 67.5 degrees, you can use the half-angle formula for sine. Since 67.5 degrees is half of 135 degrees, the formula is:

Half-Angle Formula

The half-angle formula states:

sin(θ/2) = √((1 - cos(θ)) / 2)

Applying the Formula

In this case, let θ be 135 degrees:

• First, find cos(135 degrees). Since 135 degrees is in the second quadrant, cos(135) = -√2/2.
• Now, substitute this value into the half-angle formula:

sin(67.5) = √((1 - (-√2/2)) / 2)

Simplifying the Expression

Now, simplify the expression step by step:

• 1 - (-√2/2) = 1 + √2/2
• Now, divide by 2: (1 + √2/2) / 2 = (2 + √2) / 4
• Finally, take the square root: sin(67.5) = √((2 + √2) / 4) = (√(2 + √2)) / 2.

Final Result

The exact value of sin 67.5 degrees is:

sin(67.5°) = (√(2 + √2)) / 2