Question

Find the exact value of cot theta if csc theta = -4/3 and the terminal side of theta lies in Quadrant III.

Answer 1

The exact value of cotФ is
`(√(7))/(3)`

Explanation:

Given as:

The value of cosec Ф =
`(-4)/(3)`

Let the value of cotФ = x

Now, According to question

∵ sinФ =
`(1)/(cosec\Theta )` .....1

Put the value of cosec Ф =
`(-4)/(3)` in eq 1

i.e sinФ =
`(1)/((-4)/(3) )`

Or, sinФ =
`(-3)/(4)`

Again

∵ cosФ =
`\sqrt{1-sin^(2)\Theta }`

So, cosФ =
`\sqrt{1-((-3)/(4))^(2)}`

Or, cosФ =
`\sqrt{1-((9)/(16))}`

Or, cosФ =
`\sqrt{(16 - 9)/(16))}`

∴ cosФ =
`(√(7))/(4)`

Again

we know that cotФ =
`(cos\Theta )/(sin\Theta )`

So, cotФ =
`((√(7))/(4))/((-3)/(4))`

Or, cotФ =
`(-√(7))/(3)`

As according to question sinФ lies in third quadrant

So, cotФ =
`(√(7))/(3)`

Hence, The exact value of cotФ is
`(√(7))/(3)` . Answer