Question

If 4cot theta =5 then find the value of 3sin theta + 2cos theta by 3sin theta - 2cos theta​

Answer 1

Answer:

=>cot theta = 5/4 = x/y

so r = √41

sin theta=4/√41

cos theta=5/√41

=> 3(4/√41)+2(5/√41) by 3(4/√41)-2(5√41)

=> (12+10)/(12-10)

=>22/2

=>11

ur answer is 11

Answer 2

Answer:

$\boxed{\red{\sf\:\dfrac{3\:\sin\:\theta\:+\:2\:\cos\:\theta}{3\:\sin\:\theta\:-\:2\:\cos\:\theta}\:=\:11}}$

Step-by-step-explanation:

We have given that,

$\sf\:4\:\cot\:\theta\:=\:5$

$\implies\boxed{\red{\sf\:\cot\:\theta\:=\:\dfrac{5}{4}}}$

Now, we know that,

$\pink{\sf\:\tan\:\theta\:=\:\dfrac{1}{\cot\:\theta}}\sf\:\:\:-\:-\:[\:Identity\:]$

$\implies\sf\:\tan\:\theta\:=\:\dfrac{1}{\dfrac{5}{4}}$

$\implies\boxed{\red{\sf\:\tan\:\theta\:=\:\dfrac{4}{5}}}$

Now, we know that,

$\pink{\sf\:\tan\:\theta\:=\:\dfrac{\sin\:\theta} {\cos\:\theta}}\sf\:\:\:-\:-\:[\:Identity\:]$

$\implies\sf\:\dfrac{4}{5}\:=\:\dfrac{\sin\:\theta} {\cos\:\theta}$

$\implies\sf\:\dfrac{\sin\:\theta}{\cos\:\theta}\:=\:\dfrac{4}{5}\:\:\:-\:-\:(\:1\:)$

By multiplying the numerator by 3 and denominator by 2 in equation ( 1 ), we get,

$\implies\sf\:\dfrac{3\:\times\:\sin\:\theta}{2\:\times\:\cos\:\theta}\:=\:\dfrac{3\:\times\:4}{2\:\times\:5}$

$\implies\sf\:\dfrac{3\:\sin\:\theta}{2\:\cos\:\theta}\:=\:\dfrac{12}{10}\:\:\:-\:-\:(\:2\:)$

By applying Componendo - Dividendo property in the equation ( 2 ), we get,

$\implies\sf\:\dfrac{3\:\sin\:\theta\:+\:2\:\cos\:\theta}{3\:\sin\:\theta\:-\:2\:\cos\:\theta}\:=\:\dfrac{12\:+\:10}{12\:-\:10}$

$\implies\sf\:\dfrac{3\:\sin\:\theta\:+\:2\:\cos\:\theta}{3\:\sin\:\theta\:-\:2\:\cos\:\theta}\:=\:\cancel{\dfrac{22}{2}}$

$\implies\underline{\boxed{\red{\sf\:\dfrac{3\:\sin\:\theta\:+\:2\:\cos\:\theta}{3\:\sin\:\theta\:-\:2\:\cos\:\theta}\:=\:11}}}$

$\rule{200}{1}$

Additional Information:

$\boxed{\begin{minipage}{5 cm}\underline{\bf\:Trigonometric\:Identities}\\\\\sf\:1.\:\tan\:\theta\:=\:\dfrac{\sin\:\theta}{\cos\:\theta}\\\\\sf\:2.\:\sin^2\:\theta\:+\:\cos^2\:\theta\:=\:1\\\\\sf\:3.\:1\:+\:\cot^2\:\theta\:=\:\sf\:cosec^2\:\theta\\\\\sf\:4.\:1\:+\:\tan^2\:\theta\:=\:\sec^2\:\theta\end{minipage}}$