To solve for [tex]\(\cos\left(\frac{29\pi}{6}\right)\)[/tex], we need to first rewrite [tex]\(\frac{29\pi}{6}\)[/tex] in terms of its reference angle and then use this reference angle to find the value of the cosine function.
1. Expressing the Angle Within [tex]\(0\)[/tex] to [tex]\(2\pi\)[/tex]:
[tex]\(\frac{29\pi}{6}\)[/tex] is an angle that lies beyond [tex]\(2\pi\)[/tex]. To find an equivalent angle within the range [tex]\(0\)[/tex] to [tex]\(2\pi\)[/tex], we subtract multiples of [tex]\(2\pi\)[/tex] from [tex]\(\frac{29\pi}{6}\)[/tex].
Calculate [tex]\(\left(\frac{29\pi}{6}\right) \mod \left(2\pi\right)\)[/tex]:
[tex]\[
2\pi = \frac{12\pi}{6}
\][/tex]
[tex]\[
\frac{29\pi}{6} = 4\pi + \frac{5\pi}{6}
\][/tex]
Next, determine the remainder when [tex]\(\frac{29\pi}{6}\)[/tex] is divided by [tex]\(2\pi\)[/tex]:
[tex]\[
\frac{29\pi}{6} \equiv \frac{5\pi}{6} \pmod{2\pi}
\][/tex]
So, the reference angle corresponding to [tex]\(\frac{29\pi}{6}\)[/tex] within the range [tex]\(0\)[/tex] to [tex]\(2\pi\)[/tex] is:
[tex]\[
\frac{29\pi}{6} \equiv \frac{5\pi}{6}
\][/tex]
2. Evaluating the Cosine:
Now that we have the reference angle [tex]\(\frac{5\pi}{6}\)[/tex], we can evaluate the cosine of this reference angle.
[tex]\[
\cos\left(\frac{29\pi}{6}\right) = \cos\left(\frac{5\pi}{6}\right)
\][/tex]
Recall that [tex]\(\cos\left(\frac{5\pi}{6}\right)\)[/tex] is a known value from the unit circle:
[tex]\[
\cos\left(\frac{5\pi}{6}\right) = -\frac{\sqrt{3}}{2}
\][/tex]
Therefore, the exact value of [tex]\(\cos\left(\frac{29\pi}{6}\right)\)[/tex] is:
[tex]\[
-\frac{\sqrt{3}}{2}
\][/tex]
