Fernandez Cook
06/11/2023 · High School
\( k ( w ) = \frac { 4 \cos w } { 1 - \sin w } , a = \frac { 29 \pi } { 6 } \)
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Tutor-Verified Answer
Step-by-step Solution
To analyze the function \( k(w) = \frac{4 \cos w}{1 - \sin w} \) at the point \( a = \frac{29\pi}{6} \), we can start by simplifying \( a \).
First, we can reduce \( a \) modulo \( 2\pi \):
\[
\frac{29\pi}{6} = 4\pi + \frac{5\pi}{6}
\]
Since \( 4\pi \) is equivalent to \( 0 \) in terms of periodicity (as \( 2\pi \) is a full rotation), we can simplify:
\[
\frac{29\pi}{6} \equiv \frac{5\pi}{6} \mod 2\pi
\]
Next, we will evaluate \( k(w) \) at \( w = \frac{5\pi}{6} \):
1. Calculate \( \cos\left(\frac{5\pi}{6}\right) \):
\[
\cos\left(\frac{5\pi}{6}\right) = -\frac{\sqrt{3}}{2}
\]
2. Calculate \( \sin\left(\frac{5\pi}{6}\right) \):
\[
\sin\left(\frac{5\pi}{6}\right) = \frac{1}{2}
\]
Now substitute these values into the function \( k(w) \):
\[
k\left(\frac{5\pi}{6}\right) = \frac{4 \cos\left(\frac{5\pi}{6}\right)}{1 - \sin\left(\frac{5\pi}{6}\right)} = \frac{4 \left(-\frac{\sqrt{3}}{2}\right)}{1 - \frac{1}{2}}
\]
This simplifies to:
\[
k\left(\frac{5\pi}{6}\right) = \frac{4 \left(-\frac{\sqrt{3}}{2}\right)}{\frac{1}{2}} = \frac{-2\sqrt{3}}{\frac{1}{2}} = -4\sqrt{3}
\]
Thus, the value of \( k\left(\frac{29\pi}{6}\right) \) is:
\[
\boxed{-4\sqrt{3}}
\]
Quick Answer
The value of \( k\left(\frac{29\pi}{6}\right) \) is \( -4\sqrt{3} \).
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