Question

Determine two unique values of θ in the interval [0,2π), such that sec(θ)=2√3 / 3

Answer 1

It is given that

`\sec (\theta)=\frac{2\sqrt[]{3}}{3}`

It can be written as follows:

`\sec (\theta)=\frac{2\sqrt[]{3}}{\sqrt[]{3}\sqrt[]{3}}`

`\sec (\theta)=\frac{2}{\sqrt[]{3}}`
`\text{Substitute sec}\theta=(1)/(\cos \theta)`

`(1)/(\cos \theta)=\frac{2}{\sqrt[]{3}}`

Reciprocal, we get

`\cos \theta=\frac{\sqrt[]{3}}{2}`

We know that

`\cos 30^o=\frac{\sqrt[]{3}}{2}`
`\cos (2\pi-\theta)=\cos \theta`
`\text{Substitute }\theta=30^o,\text{ we get}`

`\cos (360^o-30^o)=\cos 30^o`

`\cos 330^o=\cos 30^o=\frac{\sqrt[]{3}}{2}`

Hence we get

`\sec 330^o=\sec 30^o=\frac{2\sqrt[]{3}}{3}`