Questions: Find the exact values of the six trigonometric functions of the given angle. Do not use a calculator. 4π/3 A. cos(4π/3) = -1/2 (Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.) B. The function value is undefined. Select the correct choice below and fill in any answer boxes within your choice. A. tan(4π/3) = √3 (Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.) B. The function value is undefined. Select the correct choice below and fill in any answer boxes within your choice. A. csc(4π/3) = (Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.) B. The function value is undefined.

$Find the exact values of the six trigonometric functions of the given angle. Do not use a calculator. 4π/3 A. cos(4π/3) = -1/2 (Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.) B. The function value is undefined. Select the correct choice below and fill in any answer boxes within your choice. A. tan(4π/3) = √3 (Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.) B. The function value is undefined. Select the correct choice below and fill in any answer boxes within your choice. A. csc(4π/3) = (Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.) B. The function value is undefined.$

Transcript text: Find the exact values of the six trigonometric functions of the given angle. Do not use a calculator. \[ \frac{4 \pi}{3} \] A. $\cos \frac{4 \pi}{3}=-\frac{1}{2}$ (Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.) B. The function value is undefined. Select the correct choice below and fill in any answer boxes within your choice. A. $\tan \frac{4 \pi}{3}=\sqrt{3}$ (Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.) B. The function value is undefined. Select the correct choice below and fill in any answer boxes within your choice. A. $\csc \frac{4 \pi}{3}=$ $\square$ (Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.) B. The function value is undefined.

Solution

Solution Steps

Step 1: Calculate $\cos \frac{4 \pi}{3}$

Using the cosine addition formula: \[ \cos(a+b) = \cos(a)\cos(b) - \sin(a)\sin(b) \] we can express $\frac{4 \pi}{3}$ as $\frac{\pi}{3} + \pi$: \[ \cos\left(\frac{4 \pi}{3}\right) = \cos\left(\frac{\pi}{3} + \pi\right) = \cos\left(\frac{\pi}{3}\right)\cos(\pi) - \sin\left(\frac{\pi}{3}\right)\sin(\pi) \] Substituting the known values: \[ \cos\left(\frac{\pi}{3}\right) = \frac{1}{2}, \quad \cos(\pi) = -1, \quad \sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2}, \quad \sin(\pi) = 0 \] Thus, \[ \cos\left(\frac{4 \pi}{3}\right) = \frac{1}{2} \cdot (-1) - \frac{\sqrt{3}}{2} \cdot 0 = -\frac{1}{2} \]

Step 2: Calculate $\tan \frac{4 \pi}{3}$

Using the tangent identity: \[ \tan\left(\frac{4 \pi}{3}\right) = \frac{\sin\left(\frac{4 \pi}{3}\right)}{\cos\left(\frac{4 \pi}{3}\right)} \] We know that: \[ \sin\left(\frac{4 \pi}{3}\right) = -\sin\left(\frac{\pi}{3}\right) = -\frac{\sqrt{3}}{2} \] Thus, \[ \tan\left(\frac{4 \pi}{3}\right) = \frac{-\frac{\sqrt{3}}{2}}{-\frac{1}{2}} = \sqrt{3} \]

Step 3: Calculate $\csc \frac{4 \pi}{3}$

Using the cosecant identity: \[ \csc\left(\frac{4 \pi}{3}\right) = \frac{1}{\sin\left(\frac{4 \pi}{3}\right)} \] We already found that: \[ \sin\left(\frac{4 \pi}{3}\right) = -\frac{\sqrt{3}}{2} \] Thus, \[ \csc\left(\frac{4 \pi}{3}\right) = \frac{1}{-\frac{\sqrt{3}}{2}} = -\frac{2}{\sqrt{3}} = -\frac{2\sqrt{3}}{3} \]