Question

please solve please do​

Answer 1

$\begin{gathered}\begin{gathered}\bf \: Given \:  - \begin{cases} &\sf{3 \: sec \: \theta \: + 5 \: tan\theta \: = 8 } \end{cases}\end{gathered}\end{gathered}$

$\begin{gathered}\begin{gathered}\bf \: To\:find - \begin{cases} &\sf{5sec\theta \: + 3tan\theta \:}  \end{cases}\end{gathered}\end{gathered}$

$\begin{gathered}\Large{\bold{{\underline{Formula \: Used \::}}}} \end{gathered}$

$\rm :\longmapsto\:(1). \: {(x + y)}^{2} = {x}^{2} + {y}^{2} + 2xy$

$\rm :\longmapsto\:(2). \: {sec}^{2} x - {tan}^{2} x = 1$

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

Given that

$\rm :\longmapsto\:3sec\theta \: + 5tan\theta \: = 8$

• On squaring both sides ,we get

$\rm :\longmapsto\: {(3sec\theta \: + 5tan\theta \:)}^{2} = {8}^{2}$

$\rm :\longmapsto\: {9sec}^{2}\theta \: + {25tan}^{2}\theta \: + 30 \: sec\theta \:tan\theta \: = 64$

$\rm :\longmapsto\:9(1 + {tan}^{2}\theta) + 25( {sec}^{2}\theta - 1) + 30sec\theta \:tan\theta \: = 64$

$\rm :\longmapsto\:9 + 9 {tan}^{2}\theta \: + 25 {sec}^{2}\theta \: - 25 + 30sec\theta \:tan\theta \: = 64$

$\rm :\longmapsto\: {9tan}^{2}\theta \: + {25sec}^{2}\theta \: + 30sec\theta \:tan\theta \: = 64 + 16$

$\rm :\longmapsto\: {(3tan\theta)}^{2} + {(5sec\theta)}^{2} + 2 \times 3tan\theta \: \times 5sec\theta \: = 80$

$\rm :\longmapsto\: {(3tan\theta \: + 5sec\theta)}^{2} = 80$

$\rm :\longmapsto\:3tan\theta \: + 5sec\theta \: = \sqrt{80}$

$\rm :\longmapsto\:3tan\theta \: + 5sec\theta \: = 4 \sqrt{5}$

Additional Information:-

Relationship between sides and T ratios

sin θ = Opposite Side/Hypotenuse

cos θ = Adjacent Side/Hypotenuse

tan θ = Opposite Side/Adjacent Side

sec θ = Hypotenuse/Adjacent Side

cosec θ = Hypotenuse/Opposite Side

cot θ = Adjacent Side/Opposite Side

Reciprocal Identities

cosec θ = 1/sin θ

sec θ = 1/cos θ

cot θ = 1/tan θ

sin θ = 1/cosec θ

cos θ = 1/sec θ

tan θ = 1/cot θ

Co-function Identities

sin (90°−x) = cos x

cos (90°−x) = sin x

tan (90°−x) = cot x

cot (90°−x) = tan x

sec (90°−x) = cosec x

cosec (90°−x) = sec x

Fundamental Trigonometric Identities

sin²θ + cos²θ = 1

sec²θ - tan²θ = 1

cosec²θ - cot²θ = 1