Question
cos 5π3 = (1 point) Responses −2√2 negative the fraction with numerator square root of 2 and denominator 2 2√2 the fraction with numerator square root of 2 and denominator 2 12 1 half −3√2
Answers
Answered by
GPT-4o mini
To find \(\cos\left(\frac{5\pi}{3}\right)\), we can first determine its value by converting \(\frac{5\pi}{3}\) to degrees:
\[
\frac{5\pi}{3} \cdot \frac{180}{\pi} = 300^\circ
\]
Now, \(300^\circ\) is in the fourth quadrant. The reference angle is:
\[
360^\circ - 300^\circ = 60^\circ
\]
In the fourth quadrant, the cosine function is positive. Therefore,
\[
\cos\left(\frac{5\pi}{3}\right) = \cos(60^\circ)
\]
We know:
\[
\cos(60^\circ) = \frac{1}{2}
\]
Thus,
\[
\cos\left(\frac{5\pi}{3}\right) = \frac{1}{2}
\]
The correct response is:
\(\frac{1}{2}\) (which matches with "1 half" in your options).
\[
\frac{5\pi}{3} \cdot \frac{180}{\pi} = 300^\circ
\]
Now, \(300^\circ\) is in the fourth quadrant. The reference angle is:
\[
360^\circ - 300^\circ = 60^\circ
\]
In the fourth quadrant, the cosine function is positive. Therefore,
\[
\cos\left(\frac{5\pi}{3}\right) = \cos(60^\circ)
\]
We know:
\[
\cos(60^\circ) = \frac{1}{2}
\]
Thus,
\[
\cos\left(\frac{5\pi}{3}\right) = \frac{1}{2}
\]
The correct response is:
\(\frac{1}{2}\) (which matches with "1 half" in your options).