how you do you use the power-reducing formula to find the exact value of the trig function: sin^2(37pi/12)
To use the power-reducing formula to find the exact value of the trigonometric function, we need to express the function in terms of lower powers of the trigonometric function. The power-reducing formula for sine is:
sin^2(x) = (1 - cos(2x))/2
In this case, we want to find the exact value of sin^2(37π/12). Let's go step by step:
Step 1: Identify the angle
We have the angle 37π/12. This angle is not a common reference angle, so we need to find an equivalent angle within one full revolution (2π) to use in the power-reducing formula.
Step 2: Express the angle as a sum or difference of common reference angles
To express 37π/12 in terms of common reference angles (0, π/6, π/4, π/3, or π/2), we divide 37π by 12 to get the quotient 3 and the remainder 1. So we have:
37π/12 = 3π + π/12
Now, we can use the fact that sin(x + 2π) = sin(x) and cos(x + 2π) = cos(x) to rewrite the angle as:
37π/12 = (3π + π/12) + 2π = (3 + 1/12)π
Step 3: Apply the power-reducing formula
Using the power-reducing formula sin^2(x) = (1 - cos(2x))/2, let's calculate sin^2(37π/12):
sin^2(37π/12) = (1 - cos(2(3 + 1/12)π))/2
Now, we need the value of cos(2(3 + 1/12)π). To simplify further, we'll use the double-angle formula for cosine:
cos(2θ) = cos^2(θ) - sin^2(θ)
Let's apply the double-angle formula and replace sin^2(θ) with its appropriate expression from the power-reducing formula:
cos(2(3 + 1/12)π) = cos^2((3 + 1/12)π) - sin^2((3 + 1/12)π)
Step 4: Evaluate cosine and sine of the angle
Finally, we need to evaluate the functions at the angle (3 + 1/12)π. Substitute the angle into the cosine and sine functions, and simplify the expression.
cos^2((3 + 1/12)π) - sin^2((3 + 1/12)π)
Once simplified, this will yield the exact value of sin^2(37π/12). Make sure to compute and simplify each step carefully to obtain the correct answer.