The cosine function, (x), has a range of values between -1 and 1, inclusive. This means that for any real angle x, we must have:
This computer science problem involves algorithmic thinking and programming concepts. The solution below explains the approach, logic, and implementation step by step.
ScanSolve AI Answer
To solve the equation $\cos(2\theta + 120^\circ) = \frac{3}{2}$ for $0 < \theta < 360^\circ$:
Step 1: Analyze the range of the cosine function.
The cosine function, $\cos(x)$, has a range of values between -1 and 1, inclusive. This means that for any real angle $x$, we must have:
$$-1 \le \cos(x) \le 1$$
Step 2: Compare the given value with the range.
The equation given is $\cos(2\theta + 120^\circ) = \frac{3}{2}$.
The value $\frac{3}{2}$ is equal to 1.5.
Step 3: Determine if a solution exists.
Since $1.5 > 1$, the value $\frac{3}{2}$ falls outside the possible range of the cosine function.
Therefore, there is no real angle $x$ for which $\cos(x) = 1.5$.
Conclusion:
There are no solutions for $\theta$ that satisfy the given equation.
The final answer is $\boxed{\text{No solution}}$.
