If 0 < theta < pi and cos theta = 0.5, then what is the value of cos (2 theta)?
if cos Theta=1/2, then sinTheta=1/2 sqrt3
cos2Theta=cos^2Theta-sin^2Theta
cos2Theta= 1/4 -3/4= - 1/2
Thank you so much!
To find the value of cos(2θ), we can use the double-angle formula for cosine. The double-angle formula for cosine is given as:
cos(2θ) = cos^2(θ) - sin^2(θ)
To find the value of cos(θ), which is given as 0.5, we can use the inverse cosine function or the unit circle.
Using the inverse cosine function, we can find the value of θ:
θ = arccos(0.5)
θ ≈ 1.047
Now that we have the value of θ, we can substitute it into the double-angle formula for cosine:
cos(2θ) = cos^2(θ) - sin^2(θ)
Since 0 < θ < π, we know that cos(θ) > 0. Therefore, sin(θ) will be positive.
cos^2(θ) = (0.5)^2 = 0.25
sin^2(θ) = 1 - cos^2(θ) = 1 - 0.25 = 0.75
Substituting these values into the formula, we get:
cos(2θ) = 0.25 - 0.75
cos(2θ) = -0.5
Therefore, the value of cos(2θ) is -0.5.