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Find all solutions of the equation in the interval [tex][0, 2\pi)[/tex].

[tex]\[
2 \cos \theta - 1 = 0
\][/tex]

Write your answer in radians in terms of [tex]\pi[/tex]. If there is more than one solution, separate them with commas.

[tex]\[
\theta =
\][/tex]

[tex]\[
\square \pi
\][/tex]

Sagot :

To solve the equation [tex]\(2 \cos \theta - 1 = 0\)[/tex] within the interval [tex]\([0, 2\pi)\)[/tex], follow these steps:

1. Isolate the cosine function:

Start by solving the equation for [tex]\(\cos \theta\)[/tex].

[tex]\[ 2 \cos \theta - 1 = 0 \][/tex]

Add 1 to both sides of the equation:

[tex]\[ 2 \cos \theta = 1 \][/tex]

Divide both sides by 2:

[tex]\[ \cos \theta = \frac{1}{2} \][/tex]

2. Identify the general solutions for [tex]\(\theta\)[/tex]:

The cosine function equals [tex]\(\frac{1}{2}\)[/tex] at specific points. We know from trigonometric identities that:

[tex]\[ \cos \theta = \frac{1}{2} \quad \text{at} \quad \theta = \frac{\pi}{3} + 2k\pi \quad \text{and} \quad \theta = \frac{5\pi}{3} + 2k\pi \][/tex]

where [tex]\(k\)[/tex] is any integer.

3. Filter solutions within the interval [tex]\([0, 2\pi)\)[/tex]:

We are looking for solutions within the interval [tex]\([0, 2\pi)\)[/tex]. Let's list and check them:

- [tex]\(\theta = \frac{\pi}{3}\)[/tex]
- [tex]\(\theta = \frac{5\pi}{3}\)[/tex]

Both [tex]\(\frac{\pi}{3}\)[/tex] and [tex]\(\frac{5\pi}{3}\)[/tex] lie within the interval [tex]\([0, 2\pi)\)[/tex].

4. Write the solutions in terms of [tex]\(\pi\)[/tex]:

The solutions are:
[tex]\[ \theta = \frac{\pi}{3}, \frac{5\pi}{3} \][/tex]

So, the solutions to the equation [tex]\(2 \cos \theta - 1 = 0\)[/tex] within the interval [tex]\([0, 2\pi)\)[/tex] are:
[tex]\[ \theta = \frac{\pi}{3}, \frac{5\pi}{3} \][/tex]