Question

Find the value of sec (31π/6) without using a calculator.

Answer 1

`-(2√(3))/(3)`

Step-by-step explanation

We want to find the value of:

`\sec((31\pi)/(6))`

First, we can rewrite the angle by reducing it by 4π i.e. two full rotations of 2π to make the angle between 0 and 2π. The angle then becomes:

`\sec((7\pi)/(6))`

This is the same as:

`(1)/(\cos((7\pi)/(6)))`

Now, we can apply the reference angle by finding the equivalent value of the angle in the first quadrant:

`(1)/(\cos((7\pi)/(6)))\Rightarrow(1)/(-\cos((\pi)/(6)))`

The value is negative because the reference angle is in the third quadrant and cosine is negative in the third quadrant.

Now, solving this, we have:

`\begin{gathered} (1)/(-(√(3))/(2))\Rightarrow-(2)/(√(3)) \\ \\ -(2)/(√(3))*(√(3))/(√(3)) \\ \\ -(2√(3))/(3) \end{gathered}`

That is the answer.