Question

If sin θ =
1/2
then the value of (4 cot2

θ+4) is –​

Answer 1

Step-by-step explanation:

If sin θ =

1/2

then the value of (4 cot2

θ+4) is –

Answer 2

Question:

If $\displaystyle{\sf\:\sin\:\theta\:=\:\dfrac{1}{2}}$, then find the value of $\displaystyle{\sf\:4\:\cot^2\:\theta\:+\:4}$.

Answer:

$\displaystyle{\boxed{\red{\sf\:4\:\cot^2\:\theta\:+\:4\:=\:16}}}$

Step-by-step-explanation:

We have given that, $\displaystyle{\sf\:\sin\:\theta\:=\:\dfrac{1}{2}}$

We have to find the value of $\displaystyle{\sf\:4\:\cot^2\:\theta\:+\:4}$

Now,

$\displaystyle{\sf\:\sin\:\theta\:=\:\dfrac{1}{2}}$

$\displaystyle{\pink{\sf\:cosec\:\theta\:=\:\dfrac{1}{\sin\:\theta}}\sf\:\:\:-\:-\:-\:[\:Trigonometric\:identity\:]}$

$\displaystyle{\implies\sf\:cosec\:\theta\:=\:\dfrac{1}{\dfrac{1}{2}}}$

$\displaystyle{\implies\sf\:cosec\:\theta\:=\:\dfrac{1}{1}\:\times\:\dfrac{2}{1}}$

$\displaystyle{\implies\boxed{\red{\sf\:cosec\:\theta\:=\:2}}}$

Now, we know that,

$\displaystyle{\pink{\sf\:cosec^2\:\theta\:=\:1\:+\:\cot^2\:\theta}\sf\:\:\:-\:-\:-\:[\:Trigonometric\:identity\:]}$

$\displaystyle{\implies\sf\:(\:2\:)^2\:=\:1\:+\:\cot^2\:\theta}$

$\displaystyle{\implies\sf\:4\:=\:1\:+\:\cot^2\:\theta}$

$\displaystyle{\implies\sf\:\cot^2\:\theta\:=\:4\:-\:1}$

$\displaystyle{\implies\boxed{\red{\sf\:\cot^2\:\theta\:=\:3}}}$

Now, we have to find the value of,

$\displaystyle{\sf\:4\:\cot^2\:\theta\:+\:4}$

$\displaystyle{\implies\sf\:4\:\times\:3\:+\:4}$

$\displaystyle{\implies\sf\:12\:+\:4}$

$\displaystyle{\implies\sf\:16}$

$\displaystyle{\therefore\:\underline{\boxed{\red{\sf\:4\:\cot^2\:\theta\:+\:4\:=\:16}}}}$