Question

If 3 cot thetha= 4 then 3sin thetha +4 cos thetha / 3 sin thetha - 4 cos thetha

Answer 1

Answer:

answer:-

3cotθ−4=0

cotθ=34

Now,3sinθ−2cosθ3sinθ+4cosθ

=sinθ[3−2sinθcosθ]sinθ[3+4sinθcosθ]

=3−2cotθ3+4cotθ

=3−2×343+4×34

=3−383+316=31325

Hence, =3sinθ−2cosθ3sinθ+4cosθ=25

Answer 2

$\large\underline{\sf{Solution-}}$

$\rm :\longmapsto\:3cot\theta = 4$

$\rm\implies \:cot\theta = \dfrac{4}{3}$

Now, Consider

$\rm :\longmapsto\:\dfrac{3sin\theta + 4cos\theta }{3sin\theta - 4cos\theta }$

$\rm \:  =  \: \dfrac{sin\theta \bigg[3 + 4\dfrac{cos\theta }{sin\theta } \bigg]}{sin\theta \bigg[3 - 4\dfrac{cos\theta }{sin\theta } \bigg]}$

$\rm \:  =  \: \dfrac{3 + 4cot\theta }{3 - 4cot\theta }$

$\rm \:  =  \: \dfrac{3 + 4 \times \dfrac{4}{3} }{3 - 4 \times \dfrac{4}{3} }$

$\rm \:  =  \: \dfrac{3 + \dfrac{16}{3} }{3 - \dfrac{16}{3} }$

$\rm \:  =  \: \dfrac{\dfrac{9 + 16}{3} }{ \dfrac{9 - 16}{3} }$

$\rm \:  =  \: \dfrac{25}{ - 7}$

$\rm \:  =  \: - \: \dfrac{25}{7}$

Hence,

$\rm :\longmapsto\:\boxed{\tt{ \dfrac{3sin\theta + 4cos\theta }{3sin\theta - 4cos\theta } = - \frac{25}{7} \: }}$

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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