if cos 2theta=1/3 and 0 less than or equal to 2theta less than or equal to pi, find cos theta
since 0 <= 2θ <= π,
0 <= θ <= π/2
That is, θ is in QI.
cos2θ = 2cos^2θ-1
so cosθ = √((1+cos2θ)/2) = √((1+1/3)/2) = √(2/3)
In the first quadrant, we don't have to worry about ± signs.
To find cos(theta), we will first use the given equation cos(2theta) = 1/3.
Using the double-angle identity for cosine, we have:
cos(2theta) = 1 - 2sin^2(theta)
Substituting the given value cos(2theta) = 1/3, we get:
1/3 = 1 - 2sin^2(theta)
Rearranging the equation, we have:
2sin^2(theta) = 1 - 1/3
2sin^2(theta) = 2/3
Dividing both sides of the equation by 2, we get:
sin^2(theta) = 1/3
Taking the square root of both sides, we have:
sin(theta) = ±√(1/3)
Since 0 ≤ 2theta ≤ π, the sine value should be positive. Therefore, we have:
sin(theta) = √(1/3)
Now, we need to find cos(theta) using the Pythagorean identity:
cos^2(theta) = 1 - sin^2(theta)
Substituting the value of sin(theta) that we found, we have:
cos^2(theta) = 1 - (√(1/3))^2
cos^2(theta) = 1 - 1/3
cos^2(theta) = 2/3
Taking the square root of both sides, we get:
cos(theta) = ±√(2/3)
Since 0 ≤ 2theta ≤ π, the cosine value should be negative. Therefore, we have:
cos(theta) = -√(2/3)
To find the value of cos(theta), we'll start by using the given information that cos(2theta) = 1/3.
We can use a trigonometric identity called the double-angle identity for cosine: cos(2theta) = 2*cos^2(theta) - 1. We can rearrange this equation to solve for cos(theta).
2*cos^2(theta) - 1 = cos(2theta)
Substituting cos(2theta) = 1/3:
2*cos^2(theta) - 1 = 1/3
Now we can solve this equation for cos(theta):
2*cos^2(theta) = 1/3 + 1
2*cos^2(theta) = 4/3
Dividing both sides by 2:
cos^2(theta) = 2/3
Taking the square root of both sides:
cos(theta) = ±√(2/3)
Since 0 ≤ 2theta ≤ π, we know that 0 ≤ theta ≤ π/2 or 0 ≤ theta ≤ -π/2.
In this case, cos(theta) cannot be negative because the given range restricts theta to be positive. So, we take the positive square root:
cos(theta) = √(2/3)
Therefore, the value of cos(theta) is √(2/3).