Sure, let's go through a detailed step-by-step solution for the given equation [tex]\(3 \cos^2(x) + \sin^2(x) = 3\)[/tex].
1. Identify known trigonometric identities:
- One of the fundamental trigonometric identities is [tex]\( \sin^2(x) + \cos^2(x) = 1 \)[/tex].
2. Rewrite the equation using this identity:
- We have [tex]\(3 \cos^2(x) + \sin^2(x) = 3\)[/tex].
- From the identity [tex]\( \sin^2(x) + \cos^2(x) = 1 \)[/tex], we can express [tex]\( \sin^2(x) \)[/tex] in terms of [tex]\( \cos^2(x) \)[/tex]:
[tex]\[ \sin^2(x) = 1 - \cos^2(x) \][/tex]
3. Substitute [tex]\( \sin^2(x) \)[/tex] in the original equation:
- Replace [tex]\( \sin^2(x) \)[/tex] with [tex]\( 1 - \cos^2(x) \)[/tex] in the equation [tex]\(3 \cos^2(x) + \sin^2(x) = 3\)[/tex], giving:
[tex]\[ 3 \cos^2(x) + (1 - \cos^2(x)) = 3 \][/tex]
4. Simplify the equation:
- Combine like terms:
[tex]\[ 3 \cos^2(x) + 1 - \cos^2(x) = 3 \][/tex]
- This simplifies to:
[tex]\[ 2 \cos^2(x) + 1 = 3 \][/tex]
5. Isolate [tex]\(\cos^2(x)\)[/tex]:
- Subtract 1 from both sides of the equation:
[tex]\[ 2 \cos^2(x) = 2 \][/tex]
- Divide both sides by 2:
[tex]\[ \cos^2(x) = 1 \][/tex]
6. Determine the values of [tex]\( \cos(x) \)[/tex] that satisfy this equation:
- The equation [tex]\( \cos^2(x) = 1 \)[/tex] implies:
[tex]\[ \cos(x) = \pm 1 \][/tex]
7. Find the corresponding values of [tex]\( x \)[/tex] for [tex]\( \cos(x) = 1 \)[/tex] and [tex]\( \cos(x) = -1 \)[/tex]:
- When [tex]\( \cos(x) = 1 \)[/tex], [tex]\( x = 0, 2\pi, 4\pi, \ldots \)[/tex] (or more generally [tex]\( x = 2k\pi \)[/tex] where [tex]\( k \)[/tex] is an integer).
- When [tex]\( \cos(x) = -1 \)[/tex], [tex]\( x = \pi, 3\pi, 5\pi, \ldots \)[/tex] (or more generally [tex]\( x = (2k+1)\pi \)[/tex] where [tex]\( k \)[/tex] is an integer).
Therefore, the solutions to the equation [tex]\(3 \cos^2(x) + \sin^2(x) = 3\)[/tex] are [tex]\( x = 0, \pi, 2\pi, 3\pi, -\pi, \)[/tex] and so on.
