Final answer:
To find the exact value of sin(112.50°) * sin(112.50°) using the half-angle formula, convert the angle to radians, apply the half-angle formula, calculate cos(2*112.50°), and substitute the values into the formula.
Step-by-step explanation:
To find the exact value of the expression sin(112.50°) * sin(112.50°) using the half-angle formula, we can use the following identity:
sin²(x) = (1 - cos(2x))/2
Let's break down the steps to solve this:
- Convert the angle to radians: 112.50° = 112.50 * π/180 radians = 1.963 r
- Apply the half-angle formula: sin²(112.50°) = (1 - cos(2*112.50°))/2
- Calculate cos(2*112.50°): cos(2*112.50°) = cos(225°) = -√2/2
- Substitute the values into the formula: sin²(112.50°) = (1 - (-√2/2))/2 = (1 + √2/2)/2 = (2 + √2)/4
So, the exact value of the expression sin(112.50°) * sin(112.50°) is (2 + √2)/4.