Pregunta
Solve the trigonometric equation for all values \( 0 \leq x<2 \pi \). \[ 2 \cos x=1 \]
Ask by Sullivan Lambert. in the United States
Mar 20,2025
Solución de inteligencia artificial de Upstudy
Respuesta verificada por el tutor
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The solutions are \( x = \frac{\pi}{3} \) and \( x = \frac{5\pi}{3} \).
Solución
We start with the given equation:
\[
2 \cos x = 1
\]
Divide both sides by 2 to isolate \(\cos x\):
\[
\cos x = \frac{1}{2}
\]
We know that within one period \(0 \leq x < 2\pi\), the cosine function equals \(\frac{1}{2}\) at:
\[
x = \frac{\pi}{3} \quad \text{and} \quad x = \frac{5\pi}{3}
\]
Thus, the solutions to the equation are:
\[
x = \frac{\pi}{3},\quad \frac{5\pi}{3}
\]
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Simplifique esta solución Beyond the Answer
To solve the equation \( 2 \cos x = 1 \), first, we can simplify it by dividing both sides by 2: \[ \cos x = \frac{1}{2} \] Now, we need to find all values of \( x \) in the interval \( [0, 2\pi) \) where the cosine function equals \( \frac{1}{2} \). The cosine function is equal to \( \frac{1}{2} \) at specific standard angles: - \( x = \frac{\pi}{3} \) - \( x = \frac{5\pi}{3} \) Thus, the solutions to the equation \( 2 \cos x = 1 \) for \( 0 \leq x < 2\pi \) are: \[ x = \frac{\pi}{3}, \quad x = \frac{5\pi}{3} \]
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