Question

If 7sin^2 theta + 3cos^2theta = 4. , show that tantheta = 1/root3

Answer 1

7sin2ϴ + 3cos2ϴ = 4
4sin2 ϴ + 3sin2 ϴ+ 3cos2 ϴ = 4
4sin2 ϴ +3 (sin2 ϴ + cos2 ϴ) = 4
4sin2 ϴ + 3 = 4
4sin2 ϴ = 1 

sin2 ϴ =1/4 
sin ϴ = ½
sin ϴ =perp./hypo. = ½

So, perp. = 1k  ; hypo. = 2k
(Hypo)2  = (base)2 + (perp.)2
k2 = (base)2 + (2k)2
(base)2 = 4k2 – k2
base =  root3 k
tan ϴ = perp./base = k/root3 k
so, tan ϴ =1/root3

Answer 2

Dividing both sides by cos2θ


⇒7tan2θ+3=4sec2θ

⇒7tan2θ+3=4(1+tan2θ)

⇒7tan2θ−4tan2θ=4−3=1

⇒3tan2θ=1

⇒tanθ=±1/root3