How can I calculate the exact value of cos(1/2alpha) if alpha=arcsin1/3.
alpha=arcsin1/3 ---> sin α = 1/3 , then y = 1, r = 3
since sin α = y/r, construct a right-angled triangle using those values
x^2 + y^2 = r^2
x^2 + 1 = 9
x = ± √8 = ± 2√2
and we have cos α = ± 2√2/3
from cos 2A = 2cos α - 1
cos α = 2cos^2 (α/2) - 1
2√2/3 = 2cos^2 (α/2) - 1
2√2/3 + 1 = 2cos^2 (α/2)
2cos^2 (α/2) = (2√2 + 3)/3
cos^2 (α/2) = (2√2 + 3)/6
cos (α/2) = ± √[ (2√2 + 3)/6 ]
half-way down my solution,
from cos 2A = 2cos α - 1, should have been from cos 2A = 2cos^2 A - 1
(does not affect rest of solution), checked answer with calculator
That looks kind of messy, but note that
2(3 + 2√2) = 6+4√2 = (2+√2)^2
So, (2√2 + 3)/6 = (2+√2)^2/12
That makes
cos(α/2) = (2+√2)/(2√3) = 1/√3 + 1/√6
And a purist would still not be satisfied with 1/√3 + 1/√6 , insisting that
we rationalize the denominators
1/√3 = √3/3
1/√6 = √6/6
then 1/√3 + 1/√6 = (2√3 + √6)/6
rationalizing denominators is for Algebra I
I find it annoying and of no real use.
To calculate the exact value of cos(1/2alpha), we need to use the given value of alpha=arcsin(1/3).
Step 1: Determine the value of alpha.
Given that alpha = arcsin(1/3), we can find the value of alpha by taking the inverse sine (arcsin) of 1/3 using a calculator or math software. This will give us the angle in radians.
Step 2: Find the value of (1/2)alpha.
Now, we need to find (1/2)alpha by dividing alpha by 2.
Step 3: Calculate the value of cos(1/2alpha).
Using the value of (1/2)alpha, we can now find cos(1/2alpha). We can use the trigonometric identity cos(2θ) = 1 - 2sin^2(θ) and substitute (1/2)alpha for θ.
cos(1/2alpha) = sqrt(1 - sin^2(1/2alpha))
To solve this equation, we substitute the value of sin(1/2alpha) using the known value of alpha (from Step 1).
sin(1/2alpha) = sin(alpha/2) = sqrt(1 - cos^2(alpha/2))
Now, we have the value of cos(alpha) (from Step 1), we can substitute it into the equation.
cos(1/2alpha) = sqrt(1 - (cos(alpha/2))^2)
By substituting the values of cos(alpha/2) and cos(alpha) into the equation, we can calculate the exact value of cos(1/2alpha).
Note: Based on the information provided, we cannot calculate the exact value of cos(1/2alpha) without the numerical value of alpha (arcsin(1/3)).