Solving the Equation $\cos x = a$
📌 Why This Matters
The equation $\cos x = a$ is one of the fundamental trigonometric equations. Unlike algebraic equations that may have a finite number of solutions, trigonometric equations can have infinitely many solutions because cosine is a periodic function.
🔍 Key Properties of Cosine
- Range: $-1 \leq \cos x \leq 1$
- Period: $2\pi$ (the function repeats every $2\pi$)
- Even function: $\cos(-x) = \cos(x)$
- Key values: $\cos 0 = 1$, $\cos \frac{\pi}{2} = 0$, $\cos \pi = -1$
⚠️ Condition for solutions:
The equation $\cos x = a$ has solutions if and only if $|a| \leq 1$, i.e., $-1 \leq a \leq 1$.
If $|a| > 1$, there are no solutions.
The equation $\cos x = a$ has solutions if and only if $|a| \leq 1$, i.e., $-1 \leq a \leq 1$.
If $|a| > 1$, there are no solutions.