Question

Find the exact trigonometric ratios for the angle x whose radian measure is given. (If an answer is undefined, enter UNDEFINED.) 4π 3 sin(x) = csc(x) = cos(x) = sec(x) = tan(x) = cot(x) =

Answer 1

`\text{sin}((4\pi)/(3))=-(√(3))/(2)`

`\text{csc}((4\pi)/(3))=-(2√(3))/(3)`

`\text{cos}((4\pi)/(3))=-(1)/(2)`

`\text{sec}((4\pi)/(3))=-2`

`\text{tan}((4\pi)/(3))=√(3)`

`\text{cot}((4\pi)/(3))=(√(3))/(3)`

Explanation:

We are asked to find the exact trigonometric ratio for given angle
`x=(4\pi)/(3)`.

`\text{sin}(x)=\text{sin}((4\pi)/(3))`

`\text{sin}((4\pi)/(3))=\text{sin}(\pi+(\pi)/(3))`

Using summation identity, we will get:

`\text{sin}(\pi+(\pi)/(3))=\text{sin}(\pi)\text{cos}((\pi)/(3))+\text{cos}(\pi)\text{sin}((\pi)/(3))`

`\text{sin}(\pi+(\pi)/(3))=(0)(1)/(2)+(-1)((√(3))/(2))`

`\text{sin}(\pi+(\pi)/(3))=0+-(√(3))/(2)`

`\text{sin}((4\pi)/(3))=-(√(3))/(2)`

Let us find
`\text{csc}(x)=\text{csc}((4\pi)/(3))`

We will use identity
`\text{csc}(x)=\frac{1}{\text{sin}(x)}`

`\text{csc}((4\pi)/(3))=\frac{1}{\text{sin}((4\pi)/(3))}=(1)/(-(√(3))/(2))=-(2)/(√(3))=-(2√(3))/(3)`

Now, we will solve for cos(x).

`\text{cos}((4\pi)/(3))=\text{cos}(\pi+(\pi)/(3))`

`\text{cos}(\pi+(\pi)/(3))=\text{cos}(\pi)\text{cos}((\pi)/(3))-\text{sin}(\pi)\text{sin}((\pi)/(3))`

`\text{cos}(\pi+(\pi)/(3))=(-1)(1)/(2)-(0)((√(3))/(2))`

`\text{cos}(\pi+(\pi)/(3))=-(1)/(2)-0`

`\text{cos}((4\pi)/(3))=-(1)/(2)`

Let us find sec(x).

We will use identity
`\text{sec}(x)=\frac{1}{\text{cos}(x)}`

`\text{sec}((4\pi)/(3))=\frac{1}{\text{cos}((4\pi)/(3))}=(1)/(-(1)/(2))=-2`

Let us find tan(x).

We will use identity
`\text{tan}(x)=\frac{\text{sin}(x)}{\text{cos}(x)}`.

`\text{tan}((4\pi)/(3))=\frac{\text{sin}((4\pi)/(3))}{\text{cos}((4\pi)/(3))}`

`\text{tan}((4\pi)/(3))=(-(√(3))/(2))/(-(1)/(2))}=(√(3)*2)/(2*1)=√(3)`

Let us find cot(x).

We will use identity
`\text{cot}(x)=\frac{1}{\text{tan}(x)}`.

`\text{cot}((4\pi)/(3))=\frac{1}{\text{tan}((4\pi)/(3))}`

`\text{cot}((4\pi)/(3))=(1)/(√(3))=(√(3))/(3)`