Question #d574e

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Answer 1 · 2016-05-05T05:22:32

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

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.