cos(2theta)=(1)/(2)

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cos(2theta)=(1)/(2)

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Recognize Equation Involves Cosine: We need to solve the equation cos(2theta) = 1/2. The first step is to recognize that this equation involves the cosine of double the angle theta. We need to find the values of theta that satisfy this equation.Identify Specific Cosine Values: We know that the cosine function has a value of 1/2 at specific angles in the unit circle. These angles are 60 degrees (or pi/3 radians) and 300 degrees (or 5pi/3 radians). However, since we are dealing with cos(2theta), we need to find the angles for theta such that when doubled, they give us the angles where the cosine is 1/2.Find Theta Values: To find the values of theta, we divide the angles where cos(x) = 1/2 by 2. So, we get theta = 60 degrees / 2 = 30 degrees (or pi/3 radians / 2 = pi/6 radians) and theta = 300 degrees / 2 = 150 degrees (or 5pi/3 radians / 2 = 5pi/6 radians).Consider Periodicity: However, the cosine function is periodic with a period of 360 degrees (or 2pi radians), so we must consider all angles that are coterminal with 30 degrees and 150 degrees. This means we add k*360 degrees to each solution for theta, where k is an integer. In radians, we add k*2pi to each solution for theta.General Solutions: Therefore, the general solutions for theta are theta = 30 degrees + k*360 degrees or theta = 150 degrees + k*360 degrees, where k is an integer. In radians, theta = pi/6 + k*2pi or theta = 5pi/6 + k*2pi, where k is an integer.

$\cos(2\theta) = \frac{1}{2}$

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cos⁡(2θ)=12\cos(2\theta) = \frac{1}{2}cos(2θ)=21​

$\cos(2\theta) = \frac{1}{2}$