Question

if y = -cos x, what x-value corresponds to a y-value of \\(\frac{1}{2}\\) between \\(\pi\\) and \\(2\pi\\)? \\(\frac{?\pi}{\square}\\)

Explanation:

Step1: Set up the equation

We know that \( y = -\cos x \) and \( y=\frac{1}{2} \), so we set up the equation \( -\cos x=\frac{1}{2} \).

Step2: Solve for \(\cos x\)

Multiply both sides of the equation \( -\cos x=\frac{1}{2} \) by - 1, we get \( \cos x=-\frac{1}{2} \).

Step3: Find the angle in the given interval

We know that the cosine function \( y = \cos x \) has a value of \( -\frac{1}{2} \) at \( x=\frac{2\pi}{3}+2k\pi \) or \( x = \frac{4\pi}{3}+2k\pi \), where \( k\in\mathbb{Z} \). We are looking for the value of \( x \) in the interval \( (\pi,2\pi) \). Let's check the two general solutions:

• For \( x=\frac{2\pi}{3}+2k\pi \), when \( k = 0 \), \( x=\frac{2\pi}{3}\approx1.047\pi \) (but \( \frac{2\pi}{3}<\pi \), not in the interval). When \( k = 1 \), \( x=\frac{2\pi}{3}+2\pi=\frac{8\pi}{3}>2\pi \), not in the interval.
• For \( x=\frac{4\pi}{3}+2k\pi \), when \( k = 0 \), \( x=\frac{4\pi}{3}\approx1.333\pi \), and \( \pi<\frac{4\pi}{3}<2\pi \), which is in the interval. When \( k = 1 \), \( x=\frac{4\pi}{3}+2\pi=\frac{10\pi}{3}>2\pi \), not in the interval.

So the value of \( x \) is \( \frac{4\pi}{3} \), and if we write \( x=\frac{4}{3}\pi \), the numerator is 4 and the denominator is 3. So the fraction is \( \frac{4}{3} \) when we consider \( x = \frac{4}{3}\pi \) (i.e., the coefficient of \( \pi \) is \( \frac{4}{3} \)).

Answer:

\(\frac{4}{3}\)