To find all values of [tex]\( x \)[/tex] in the interval [tex]\( 0 \leq x \leq 2\pi \)[/tex] for which [tex]\( \cos x = 0 \)[/tex], we can follow these steps:
1. Understand the Problem: We are looking for the values of [tex]\( x \)[/tex] where the cosine function evaluates to zero within the given range.
2. Recall the Properties of the Cosine Function: The cosine function, [tex]\( \cos x \)[/tex], oscillates between [tex]\(-1\)[/tex] and [tex]\(1\)[/tex] and has zeros at specific points within any interval of [tex]\( 2\pi \)[/tex].
3. Identify the Zeroes of Cosine within One Period: Within one period of [tex]\( 2\pi \)[/tex], the cosine function is zero at [tex]\( \frac{\pi}{2} \)[/tex] and [tex]\( \frac{3\pi}{2} \)[/tex]. This is because cosine plots a complete wave every [tex]\( 2\pi \)[/tex], transitioning from 1 to -1 and back to 1, crossing the x-axis (where cosine equals 0) at these points:
- [tex]\( \frac{\pi}{2} \)[/tex]
- [tex]\( \frac{3\pi}{2} \)[/tex]
4. Confirm the Zeros within the Interval [0, 2π]: The points [tex]\( \frac{\pi}{2} \)[/tex] and [tex]\( \frac{3\pi}{2} \)[/tex] both fall within the interval [tex]\(0 \leq x \leq 2\pi\)[/tex].
Therefore, the values of [tex]\( x \)[/tex] that satisfy [tex]\( \cos x = 0 \)[/tex] within the interval [tex]\( 0 \leq x \leq 2\pi \)[/tex] are:
[tex]\[ x = \frac{\pi}{2}, \frac{3\pi}{2}. \][/tex]
These are the only points in the given interval where the cosine function evaluates to zero.
