Find the following trigonometric values. Express your answers exactly. {:[cos((4pi)/(3))=◻],[sin((4pi)/(3))=◻]:}

Determine Quadrant: Determine the quadrant in which the angle (4pi)/3 lies. The angle (4pi)/3 is greater than pi but less than 3pi/2, which places it in the third quadrant of the unit circle.Properties of Cosine and Sine: Recall the properties of cosine and sine in the third quadrant. In the third quadrant, both cosine and sine are negative.Find Reference Angle: Find the reference angle for (4pi)/3. The reference angle is the acute angle that the terminal side of (4pi)/3 makes with the x-axis. Since (4pi)/3 is pi/3 more than pi, the reference angle is pi - (4pi)/3 = pi/3.Use Reference Angle: Use the reference angle to find the exact values of cosine and sine. Since the reference angle is pi/3, we know that cos(pi/3) = 1/2 and sin(pi/3) = sqrt(3)/2. However, because (4pi)/3 is in the third quadrant, we must take the negative of these values.Write Final Answers: Write the final answers. cos((4pi)/3) = -cos(pi/3) = -1/2 sin((4pi)/3) = -sin(pi/3) = -sqrt(3)/2

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Find the following trigonometric values.
Express your answers exactly.

{:[cos((4pi)/(3))=◻],[sin((4pi)/(3))=◻]:}

Find the following trigonometric values. $\newline$ Express your answers exactly. $\newline$ $\begin{array}{l} \cos \left(\frac{4 \pi}{3}\right)=\square \\ \sin \left(\frac{4 \pi}{3}\right)=\square \end{array}$

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Q. Find the following trigonometric values.

\newline

Express your answers exactly.

\newline

\begin{array}{l} \cos \left(\frac{4 \pi}{3}\right)=\square \\ \sin \left(\frac{4 \pi}{3}\right)=\square \end{array}

Determine Quadrant: Determine the quadrant in which the angle $(4\pi)/3$ lies. $\newline$ The angle $(4\pi)/3$ is greater than $\pi$ but less than $3\pi/2$ , which places it in the third quadrant of the unit circle.
Properties of Cosine and Sine: Recall the properties of cosine and sine in the third quadrant. In the third quadrant, both cosine and sine are negative.
Find Reference Angle: Find the reference angle for $(4\pi)/3$ . $\newline$ The reference angle is the acute angle that the terminal side of $(4\pi)/3$ makes with the x-axis. Since $(4\pi)/3$ is $\pi/3$ more than $\pi$ , the reference angle is $\pi - (4\pi)/3 = \pi/3$ .
Use Reference Angle: Use the reference angle to find the exact values of cosine and sine. $\newline$ Since the reference angle is $\pi/3$ , we know that $\cos(\pi/3) = 1/2$ and $\sin(\pi/3) = \sqrt{3}/2$ . However, because $(4\pi)/3$ is in the third quadrant, we must take the negative of these values.
Write Final Answers: Write the final answers. $\newline$ $\cos\left(\frac{4\pi}{3}\right) = -\cos\left(\frac{\pi}{3}\right) = -\frac{1}{2}$ $\newline$ $\sin\left(\frac{4\pi}{3}\right) = -\sin\left(\frac{\pi}{3}\right) = -\frac{\sqrt{3}}{2}$