Question

Find the exact values of the six trigonometric functions of the given angle. If any are not defined, say "not defined." Do not use a calculator.

$\frac{2\pi }{3}$

Short answer

The exact values of the six trigonometric functions are$\sin \left(\frac{2\pi }{3}\right)=\frac{\sqrt{3}}{2}$,$\cos \left(\frac{2\pi }{3}\right)=-\frac{1}{2}$,$\tan \left(\frac{2\pi }{3}\right)=-\sqrt{3}$,$\csc \left(\frac{2\pi }{3}\right)=\frac{2}{\sqrt{3}}$,$\sec \left(\frac{2\pi }{3}\right)=-2$and$\cot \left(\frac{2\pi }{3}\right)=-\frac{1}{\sqrt{3}}$.

Step-by-step solution

Step 1. Determine a point corresponds to the given angle.

The given angle is $\frac{2\pi }{3}$. This angle is multiple of $\frac{\pi }{3}$.

From the below figure, we see the point $\left(-\frac{1}{2},\frac{\sqrt{3}}{2}\right)$corresponds to $\frac{2\pi }{3}$.

Step 2. Determine the exact value of sine function.

The exact value of the sine function is:

$$\begin{gathered} \sin \left(\frac{2\pi }{3}\right)=y \\ \sin \left(\frac{2\pi }{3}\right)=\frac{\sqrt{3}}{2} \end{gathered}$$

Step 3. Determine the exact value of cosine function.

The exact value of the cosine function is:

$$\begin{gathered} \cos \left(\frac{2\pi }{3}\right)=x \\ \cos \left(\frac{2\pi }{3}\right)=-\frac{1}{2} \end{gathered}$$

Step 4. Determine the exact value of tangent function.

The exact value of the tangent function is:

$\begin{array}{rcl}\tan \theta & =& \frac{\sin \theta }{\cos \theta }\\ \tan \left(\frac{2\pi }{3}\right)& =& \frac{\frac{\sqrt{3}}{2}}{-\frac{1}{2}}\\ \tan \left(\frac{2\pi }{3}\right)& =& -\sqrt{3}\end{array}$

Step 5. Determine the exact value of cosecant function.

The exact value of the cosecant function is:

$\begin{array}{rcl}\csc \theta & =& \frac{1}{\sin \theta }\\ \csc \left(\frac{2\pi }{3}\right)& =& \frac{1}{\frac{\sqrt{3}}{2}}\\ \csc \left(\frac{2\pi }{3}\right)& =& \frac{2}{\sqrt{3}}\end{array}$

Step 6. Determine the exact value of secant function.

The exact value of the secant function is:

$\begin{array}{rcl}\sec \theta & =& \frac{1}{\cos \theta }\\ \sec \left(\frac{2\pi }{3}\right)& =& \frac{1}{\frac{-1}{2}}\\ \sec \left(\frac{2\pi }{3}\right)& =& -2\end{array}$

Step 7. Determine the exact value of cotangent function.

The exact value of the cotangent function is:

$\begin{array}{rcl}\cot \theta & =& \frac{1}{\tan \theta }\\ \cot \left(\frac{2\pi }{3}\right)& =& \frac{1}{-\sqrt{3}}\\ \cot \left(\frac{2\pi }{3}\right)& =& -\frac{1}{\sqrt{3}}\end{array}$