Final answer:
To solve the equation cos (θ/2) − 1 = 0, find the angles where the cosine value is equal to 1. This gives the general solution θ = 4kπ, where k is any integer.
Step-by-step explanation:
The equation in question is cos (θ/2) − 1 = 0. To solve it, we set the cosine term equal to 1 since the equation then becomes cos (θ/2) = 1. This occurs when the argument of the cosine function is an even multiple of π, which are the angles where the cosine function has a value of 1. Thus, θ/2 = 2kπ, where k is any integer. To find θ, we multiply both sides by 2 to obtain θ = 4kπ. The solutions are all the angles that satisfy this equation, given in radians.
When solving trigonometric equations, it's important to consider the periodicity of the sine and cosine functions. Here, the cosine function has a period of 2π, which is how the unknowns are represented in terms of periodic intervals.
Therefore, the set of all solutions to the equation cos (θ/2) − 1 = 0 is given by θ = 4kπ, where k is any integer.