Use the half-angle formulas to come up with an exact expression for each function value below. You do not have to simplify your answers.
cos(pi/16)
To find an exact expression for cos(pi/16) using the half-angle formula, we start with the following formula:
cos(θ/2) = ±√((1 + cos(θ))/2)
In this case, θ = π/8. Substituting this into the formula, we have:
cos(π/16) = ±√((1 + cos(π/8))/2)
Since we have the positive value of θ/2, we can simplify further:
cos(π/16) = √((1 + cos(π/8))/2)
Therefore, an exact expression for cos(π/16) using the half-angle formula is √((1 + cos(π/8))/2).
To find an exact expression for cos(pi/16) using the half-angle formulas, we can start by using the formula for cos(θ/2):
cos(θ/2) = ±√((1 + cos(θ))/2)
In this case, θ = pi/8, so we can rewrite the formula as:
cos(pi/16) = ±√((1 + cos(pi/8))/2)
Now, to find the exact expression, we need to find the value of cos(pi/8). We can use the half-angle formula again, this time with θ = pi/4:
cos(pi/4) = ±√((1 + cos(pi/2))/2)
Since we know the values of cos(pi/2) = 0 and cos(pi/4) = ±√(2)/2, we can substitute them into the formula:
cos(pi/8) = ±√((1 + (√(2)/2))/2)
Simplifying the expression inside the square root:
cos(pi/8) = ±√((2 + √(2))/4)
Finally, substitute this expression back into the original formula for cos(pi/16):
cos(pi/16) = ±√((1 + (√((2 + √(2))/4)))/2)
And there you have it! An exact expression for cos(pi/16) using the half-angle formulas.
cos (x/2) = sqrt[(1/2)(1 + cosx)]
cos(pi/16) = sqrt[(1/2)(1 + cos(pi/8)]
cos(pi/8) = sqrt[(1/2)(1 + cos(pi/4)]
= sqrt[(1/2)(1 + sqrt2/2] = 0.92388..
cos(pi/16) = sqrt[(1/2)*1.92388]
= 0.980785..
Writing it in exact form would be a bit messy, but I hope you see how it can be done