Math, asked by Ayush7810, 1 month ago

if 3sin0+4 cos0=5 , find sin0​

Answers

Answered by mathdude500
0

\large\underline{\bold{Given \:Question - }}

 \sf \: If 3sin\theta \: + 4cos\theta \: = 5, \: find \:sin\theta \:

\large\underline{\bold{Solution-}}

Identities Used :-

  1. \:  \:  \: \boxed{ \bf{ {(x + y)}^{2}  =  {x}^{2} + 2xy +  {y}^{2}  }}

  2. \:  \:  \: \boxed{ \bf{ {sin}^{2}\theta \: +  {cos}^{2}\theta \: = 1  }}

  3. \:  \:  \: \boxed{ \bf{ {(x - y)}^{2}  =  {x}^{2} - 2xy +  {y}^{2}  }}

Let solve the problem now!!

Given that

 \sf \:  3sin\theta \: + 4cos\theta \: = 5

 \sf \:4cos\theta \: = 5 - 3sin\theta \:

On squaring both sides, we get

 \sf \:  {(4cos\theta \:)}^{2}  =  {(5 - 3sin\theta \:)}^{2}

 \sf \:  16 {cos}^{2} \theta \: = 25 +  {9sin}^{2} \theta \: - 30sin\theta \:

 \sf \: 16(1 -  {sin}^{2}\theta \:) = 25 + 9 {sin}^{2}  \theta \: - 30sin\theta \:

 \sf \: 16 - 16 {sin}^{2} \theta \: = 25 +  {9sin}^{2} \theta \: - 30sin\theta \:

 \sf \: 25 {sin}^{2} \theta \: - 30sin\theta \:  + 9 = 0

 \sf \:  {(5sin\theta \:)}^{2}  +  {(3)}^{2}  - 2 \times 5sin\theta \: \times 3 = 0

 \sf \:  {(5sin\theta \: - 3)}^{2}  = 0

 \sf \:  \: 5sin\theta \: - 3 = 0

   \:  \:  \:  \: \therefore \boxed{\bf \: sin\theta \: = \dfrac{3}{5} }

Additional Information :-

  1. \:  \:  \: \boxed{ \bf{ {sec}^{2}\theta \: -  {tan}^{2}\theta \: = 1  }}

  2. \:  \:  \: \boxed{ \bf{ {cosec}^{2}\theta \: -  {cot}^{2}\theta \: = 1  }}

  3. \:  \:  \: \boxed{ \bf{cosec\theta \: - cot\theta \: = \dfrac{1}{cosec\theta \: + cot\theta \:} }}

  4. \:  \:  \: \boxed{ \bf{cosec\theta \: + cot \theta \:= \dfrac{1}{cosec\theta \: - cot\theta \:} }}

  5. \:  \:  \: \boxed{ \bf{sec\theta \: + tan\theta \: = \dfrac{1}{ sec\theta \: - tan\theta \: } }}

  6. \:  \:  \: \boxed{ \bf{sec\theta \:  -  tan\theta \: = \dfrac{1}{ sec\theta \:  +  tan\theta \: } }}

Similar questions