Question

Question #d574e

Answer 1

`x=45^@`is the correct value of `x`
If at `x=26.56505118^@` then `tan x=1/2`

Explanation:

Solve the trigonometric equation

`1+sin^2 x=3 sin x cos x`

Let `cos x=sqrt(1-sin^2 x)" "`from Pythagorean Relation

`1+sin^2 x=3 sin x sqrt(1-sin^2 x)`

Square both sides of the equation

`1+sin^2 x=3 sin x sqrt(1-sin^2 x)`

`(1+sin^2 x)^2=(3 sin x sqrt(1-sin^2 x))^2`

`1+2*sin^2 x+sin^4 x=9*sin^2 x(1-sin^2 x)`

`1+2*sin^2 x+sin^4 x=9*sin^2 x-9*sin^4 x`

`10*sin^4 x-7*sin^2 x+1=0`

Quadratic Equation

Let `w^2=sin^4 x` and `w=sin^2 x` then we have

`10*sin^4 x-7*sin^2 x+1=0`

`10*w^2-7*w+1=0`

then solve for w

`w=(-b+-sqrt(b^2-4ac))/(2a)`

`w=(-(-7)+-sqrt((-7)^2-4(10)(1)))/(2(10))`

`w=(7+-sqrt(49-40))/(2(10))`

`w_1=(7+3)/20=1/2`
`w_2=(7-3)/20=1/5`

`sin^2 x=+-sqrt(1/2)` and

`x=45^@` which is the root of the equation

`x=-45^@` not a root
`x=+-26.565^@` not a root

God bless...I hope the explanation is useful.