Question

use a half-angle formula to find the exact value of the following expression. sin 112.50 sin 112.50 (simplify your answer, including any radicals. use integers or fractions for any numbers in the expressiom rationalize the denominator.)

Answer 1

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:

1. Convert the angle to radians: 112.50° = 112.50 * π/180 radians = 1.963 r
2. Apply the half-angle formula: sin²(112.50°) = (1 - cos(2*112.50°))/2
3. Calculate cos(2*112.50°): cos(2*112.50°) = cos(225°) = -√2/2
4. 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.