Question

Use the unit circle to find sec(7/6)

Answer 1

Step 1

Draw the unit circle required

Step 2

Find the value sec(7π/6) in cosine

`\begin{gathered} \sec ((7\pi)/(6))=(1)/(cos((7\pi)/(6))) \\ \sec (x)=(1)/(cos(x)) \end{gathered}`

Step 3

Find cos(7π/6)

The trigonometric unit circle and a trigonometric table gives;

`\begin{gathered} \cos ((7\pi)/(6))=\cos ((\pi)/(6)+\pi) \\ \cos ((7\pi)/(6))=\text{cos}((\pi)/(6))\cos (\pi)-\sin ((\pi)/(6))sin\pi=-\cos ((\pi)/(6)) \\ \cos ((7\pi)/(6))=\frac{\sqrt[]{3}}{2}(-1)-((1)/(2))(0)=-\frac{\sqrt[]{3}}{2} \\ \cos ((7\pi)/(6))=-\frac{\sqrt[]{3}}{2} \end{gathered}`

Step 4

Find sec(7π/6)

`\begin{gathered} \sec (x)=(1)/(cos(x)) \\ \text{sec}((7\pi)/(6))=(1)/(\cos ((7\pi)/(6))) \\ \text{sec}((7\pi)/(6))=\frac{1}{-\frac{\sqrt[]{3}}{2}} \\ \text{sec}((7\pi)/(6))=-\frac{2}{\sqrt[]{3}} \end{gathered}`

Step 5

Rationalize the denominator

`\begin{gathered} \sec ((7\pi)/(6))=-\frac{2}{\sqrt[]{3}}*\frac{\sqrt[]{3}}{\sqrt[]{3}} \\ \sec ((7\pi)/(6))=-\frac{2\sqrt[]{3}}{\sqrt[]{9}} \\ \sec ((7\pi)/(6))=-\frac{2\sqrt[]{3}}{3} \end{gathered}`

Hence,

`\sec ((7\pi)/(6))=-\frac{2\sqrt[]{3}}{3}`