Question

plzzzzz solve it.... ​

Answer 1

Answer:-

$\displaystyle{\boxed{\red{\sf\:\dfrac{5\:\sin\:\theta\:-\:3\:\cos\:\theta}{5\:\sin\:\theta\:+\:3\:\cos\:\theta}\:=\:\dfrac{1}{7}}}}$

Step-by-step-explanation:-

We have given that $\displaystyle{\sf\:5\:\tan\:\theta\:=\:4}$.

We have to find the value of $\displaystyle{\sf\:\dfrac{5\:\sin\:\theta\:-\:3\:\cos\:\theta}{5\:\sin\:\theta\:+\:3\:\cos\:\theta}}$.

$\displaystyle{\sf\:5\:\tan\:\theta\:=\:4\:\:\:-\:-\:[\:Given\:]}$

$\displaystyle{\implies\sf\:5\:\times\:\dfrac{\sin\:\theta}{\cos\:\theta}\:=\:4\:\:\:-\:-\:\left[\:\because\:\tan\:\theta\:=\:\dfrac{\sin\:\theta}{\cos\:\theta}\:\right]}$

$\displaystyle{\implies\sf\:\dfrac{5\:\sin\:\theta}{\cos\:\theta}\:\times\:\dfrac{1}{3}\:=\:\dfrac{4}{1}\:\times\:\dfrac{1}{3}\:\:\:-\:-\:-\:\left[\:Multiplying\:both\:sides\:by\:\dfrac{1}{3}\:\right]}$

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

$\displaystyle{\implies\sf\:\dfrac{5\:\sin\:\theta\:+\:3\:\cos\:\theta}{5\:\sin\:\theta\:-\:3\:\cos\:\theta}\:=\:\dfrac{4\:+\:3}{4\:-\:3}\:\:\:-\:-\:[\:By\:using\:Componendo\:-\:dividendo\:]}$

$\displaystyle{\implies\sf\:\dfrac{5\:\sin\:\theta\:+\:3\:\cos\:\theta}{5\:\sin\:\theta\:-\:3\:\cos\:\theta}\:=\:\dfrac{7}{1}}$

$\displaystyle{\implies\sf\:\dfrac{5\:\sin\:\theta\:+\:3\cos\:\theta}{5\:\sin\:\theta\:-\:3\:\cos\:\theta}\:=\:7}$

$\displaystyle{\implies\boxed{\red{\sf\:\dfrac{5\:\sin\:\theta\:-\:3\:\cos\:\theta}{5\:\sin\:\theta\:+\:3\:\cos\:\theta}\:=\:\dfrac{1}{7}}}\sf\:\:\:-\:-\:[\:By\:using\:Invertendo\:]}$

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Additional Information:—

$\displaystyle{\boxed{\begin{minipage}{4.5 cm} \underline{\bf\:Trigonometric\:Identities}\\\\\sf\:1.\:\tan\:\theta\:=\:\dfrac{\sin\:\theta}{\cos\:\theta}\\\\\sf\:2.\:\sin^2\:\theta\:+\:\cos^2\:\theta\:=\:1\\\\\sf\:3.\:1\:+\:\cot^2\:\theta\:=\:\sf\:cosec^2\:\theta\\\\\sf\:4.\:1\:+\:\tan^2\:\theta\:=\:\sec^2\:\theta\end{minipage}}}$

Answer 2

Answer:

$\displaystyle{\sf\:5\:\tan\:\theta\:=\:4\:\:\:-\:-\:[\:Given\:]}$

$\displaystyle{\implies\sf\:5\:\times\:\dfrac{\sin\:\theta}{\cos\:\theta}\:=\:4\:\:\:-\:-\:\left[\:\because\:\tan\:\theta\:=\:\dfrac{\sin\:\theta}{\cos\:\theta}\:\right]}$

$\displaystyle{\implies\sf\:\dfrac{5\:\sin\:\theta}{\cos\:\theta}\:\times\:\dfrac{1}{3}\:=\:\dfrac{4}{1}\:\times\:\dfrac{1}{3}\:\:\:-\:-\:-\:\left[\:Multiplying\:both\:sides\:by\:\dfrac{1}{3}\:\right]}$

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

$\displaystyle{\implies\sf\:\dfrac{5\:\sin\:\theta\:+\:3\:\cos\:\theta}{5\:\sin\:\theta\:-\:3\:\cos\:\theta}\:=\:\dfrac{4\:+\:3}{4\:-\:3}\:\:\:-\:-\:[\:By\:using\:Componendo\:-\:dividendo\:]}$

$\displaystyle{\implies\sf\:\dfrac{5\:\sin\:\theta\:+\:3\:\cos\:\theta}{5\:\sin\:\theta\:-\:3\:\cos\:\theta}\:=\:\dfrac{7}{1}}$

$\displaystyle{\implies\sf\:\dfrac{5\:\sin\:\theta\:+\:3\cos\:\theta}{5\:\sin\:\theta\:-\:3\:\cos\:\theta}\:=\:7}$

$\displaystyle{\implies\boxed{\red{\sf\:\dfrac{5\:\sin\:\theta\:-\:3\:\cos\:\theta}{5\:\sin\:\theta\:+\:3\:\cos\:\theta}\:=\:\dfrac{1}{7}}}\sf\:\:\:-\:-\:[\:By\:using\:Invertendo\:]}$

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